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# Sestems of linear equations

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### Sestems of linear equations

1. 1. miL2872X_ch03_177-254 09:22:2006 02:15 PM IA Page 177 CONFIRMING PAGES Systems of Linear Equations 3 3.1 Solving Systems of Linear Equations by Graphing 3.2 Solving Systems of Equations by Using the Substitution Method 3.3 Solving Systems of Equations by Using the Addition Method 3.4 Applications of Systems of Linear Equations in Two Variables 3.5 Systems of Linear Equations in Three Variables and Applications 3.6 Solving Systems of Linear Equations by Using Matrices 3.7 Determinants and Cramer’s Rule In this chapter we solve systems of linear equations in two and three variables. Some new terms are introduced in the first section of this chapter. Unscramble each word to find a key word from this chapter. As a hint, there is a clue for each word. Complete the word scramble to familiarize yourself with the key terms. 1. NNEDPNTDIEE 2. NNCIEOSTINST 3. LOUINSTO 4. INENSCOSTT 5. DNEEDPETN (A system of two linear equations representing more than one line) (A system having no solution) (An ordered pair that satisfies both equations in a system of two equations) (A system of equations that has one or more solutions) (A system of two linear equations that represents only one line) 177
2. 2. miL2872X_ch03_177-254 09:22:2006 178 02:15 PM IA Page 178 CONFIRMING PAGES Chapter 3 Systems of Linear Equations Section 3.1 Concepts 1. Solutions to Systems of Linear Equations 2. Dependent and Inconsistent Systems of Linear Equations 3. Solving Systems of Linear Equations by Graphing Solving Systems of Linear Equations by Graphing 1. Solutions to Systems of Linear Equations A linear equation in two variables has an infinite number of solutions that form a line in a rectangular coordinate system. Two or more linear equations form a system of linear equations. For example: x Ϫ 3y ϭ Ϫ5 2x ϩ 4y ϭ 10 A solution to a system of linear equations is an ordered pair that is a solution to each individual linear equation. Example 1 Determining Solutions to a System of Linear Equations Determine whether the ordered pairs are solutions to the system. x ϩ y ϭ Ϫ6 a. 1Ϫ2, Ϫ42 b. 10, Ϫ62 3x Ϫ y ϭ Ϫ2 Solution: a. Substitute the ordered pair 1Ϫ2, Ϫ42 into both equations: 1Ϫ22 ϩ 1Ϫ42 ϭ Ϫ6 ✔ True x ϩ y ϭ Ϫ6 31Ϫ22 Ϫ 1Ϫ42 ϭ Ϫ2 ✔ True 3x Ϫ y ϭ Ϫ2 Because the ordered pair 1Ϫ2, Ϫ42 is a solution to both equations, it is a solution to the system of equations. b. Substitute the ordered pair 10, Ϫ62 into both equations: 102 ϩ 1Ϫ62 ϭ Ϫ6 ✔ True x ϩ y ϭ Ϫ6 3102 Ϫ 1Ϫ62 ՘ Ϫ2 3x Ϫ y ϭ Ϫ2 False Because the ordered pair 10, Ϫ62 is not a solution to the second equation, it is not a solution to the system of equations. Skill Practice 1. Determine whether the ordered pairs are solutions to the system. 3x ϩ 2y ϭ Ϫ8 y ϭ 2x Ϫ 18 a. (Ϫ2, Ϫ1) Skill Practice Answers 1a. No b. Yes b. (4, Ϫ10)
3. 3. miL2872X_ch03_177-254 09:22:2006 02:15 PM IA Page 179 Section 3.1 CONFIRMING PAGES Solving Systems of Linear Equations by Graphing y A solution to a system of two linear equations may be interpreted graphically as a point of intersection between the two lines. Notice that the lines intersect at 1Ϫ2, Ϫ42 (Figure 3-1). 5 4 3 3x Ϫ y ϭ Ϫ2 2 1 Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 Ϫ1 Ϫ2 (Ϫ2, Ϫ4) 1 2 3 4 5 x Ϫ3 Ϫ4 Ϫ5 x ϩ y ϭ Ϫ6 Figure 3-1 2. Dependent and Inconsistent Systems of Linear Equations When two lines are drawn in a rectangular coordinate system, three geometric relationships are possible: 1. Two lines may intersect at exactly one point. 2. Two lines may intersect at no point. This occurs if the lines are parallel. 3. Two lines may intersect at infinitely many points along the line. This occurs if the equations represent the same line (the lines are coinciding). If a system of linear equations has one or more solutions, the system is said to be a consistent system. If a linear equation has no solution, it is said to be an inconsistent system. If two equations represent the same line, then all points along the line are solutions to the system of equations. In such a case, the system is characterized as a dependent system. An independent system is one in which the two equations represent different lines. Solutions to Systems of Linear Equations in Two Variables One unique solution No solution Infinitely many solutions One point of intersection Parallel lines Coinciding lines System is consistent. System is independent. System is inconsistent. System is independent. System is consistent. System is dependent. 179
4. 4. miL2872X_ch03_177-254 09:22:2006 180 02:15 PM IA Page 180 CONFIRMING PAGES Chapter 3 Systems of Linear Equations 3. Solving Systems of Linear Equations by Graphing Example 2 Solving a System of Linear Equations by Graphing Solve the system by graphing both linear equations and finding the point(s) of intersection. 3 1 yϭϪ xϩ 2 2 2x ϩ 3y ϭ Ϫ6 Solution: To graph each equation, write the equation in slope-intercept form y ϭ mx ϩ b. First equation: 3 1 yϭϪ xϩ 2 2 Second equation: Slope: Ϫ3 2 2x ϩ 3y ϭ Ϫ6 3y ϭ Ϫ2x Ϫ 6 3y Ϫ2x 6 ϭ Ϫ 3 3 3 2 yϭϪ xϪ2 3 Slope: Ϫ2 3 From their slope-intercept forms, we see that the lines have different slopes, indicating that the lines must intersect at exactly one point. Using the slope and y-intercept we can graph the lines to find the point of intersection (Figure 3-2). y 5 4 3 2 1 Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 Ϫ1 2x ϩ 3y ϭ Ϫ6 Ϫ2 y ϭ Ϫ 3x ϩ 2 1 2 Ϫ3 Ϫ4 Ϫ5 3 4 1 2 5 x Point of intersection (3, Ϫ4) Figure 3-2 The point 13, Ϫ42 appears to be the point of intersection. This can be confirmed by substituting x ϭ 3 and y ϭ Ϫ4 into both equations. 3 1 yϭϪ xϩ 2 2 2. y 2 1 Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 Ϫ1 Ϫ2 2132 ϩ 31Ϫ42 ՘ Ϫ6 6 Ϫ 12 ϭ Ϫ6 ✔ True The solution is 13, Ϫ42. 5 4 3 (Ϫ2, 1) 1 9 Ϫ4 ϭ Ϫ ϩ ✔ True 2 2 2x ϩ 3y ϭ Ϫ6 Skill Practice Answers 1 3 Ϫ4 ՘ Ϫ 132 ϩ 2 2 Skill Practice 1 2 3 4 5 x 2. Solve by using the graphing method. Ϫ3 3x ϩ y ϭ Ϫ5 Ϫ4 Ϫ5 x Ϫ 2y ϭ Ϫ4
5. 5. miL2872X_ch03_177-254 09:22:2006 02:15 PM IA Page 181 Section 3.1 CONFIRMING PAGES 181 Solving Systems of Linear Equations by Graphing TIP: In Example 2, the lines could also have been graphed by using the x- and y-intercepts or by using a table of points. However, the advantage of writing the equations in slope-intercept form is that we can compare the slopes and y-intercepts of each line. 1. If the slopes differ, the lines are different and nonparallel and must cross in exactly one point. 2. If the slopes are the same and the y-intercepts are different, the lines are parallel and do not intersect. 3. If the slopes are the same and the y-intercepts are the same, the two equations represent the same line. Example 3 Solving a System of Linear Equations by Graphing Solve the system by graphing. 4x ϭ 8 6y ϭ Ϫ3x ϩ 6 Solution: The first equation 4x ϭ 8 can be written as x ϭ 2. This is an equation of a vertical line. To graph the second equation, write the equation in slopeintercept form. First equation: y Second equation: 4x ϭ 8 6y ϭ Ϫ3x ϩ 6 xϭ2 6y Ϫ3x 6 ϭ ϩ 6 6 6 5 4 3 1 yϭϪ xϩ1 2 4x ϭ 8 2 6y ϭ Ϫ3x ϩ 6 1 Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 Ϫ1 Ϫ2 (2, 0) 1 2 3 4 5 x Ϫ3 Ϫ4 Ϫ5 Figure 3-3 The graphs of the lines are shown in Figure 3-3. The point of intersection is (2, 0). This can be confirmed by substituting (2, 0) into both equations. 4x ϭ 8 4122 ϭ 8 ✓ True 6y ϭ Ϫ3x ϩ 6 6102 ϭ Ϫ3122 ϩ 6 ✓ True The solution is (2, 0). Skill Practice Answers Skill Practice 3. 3. Solve the system by graphing. Ϫ4 ϭ Ϫ4y Ϫ3x Ϫ y ϭ Ϫ4 y 5 4 3 2 1 Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 Ϫ1 Ϫ2 Ϫ3 Ϫ4 Ϫ5 (1, 1) 1 2 3 4 5 x
6. 6. miL2872X_ch03_177-254 09:22:2006 182 02:15 PM IA Page 182 CONFIRMING PAGES Chapter 3 Systems of Linear Equations Example 4 Solving a System of Equations by Graphing Solve the system by graphing. Ϫx ϩ 3y ϭ Ϫ6 6y ϭ 2x ϩ 6 Solution: To graph the line, write each equation in slope-intercept form. y First equation: 5 4 3 y ϭ1x ϩ 1 3 2 1 Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 Ϫ1 Ϫ2 1 2 3 4 x 5 y ϭ1x Ϫ 2 3 Second equation: Ϫx ϩ 3y ϭ Ϫ6 6y ϭ 2x ϩ 6 3y ϭ x Ϫ 6 3y x 6 ϭ Ϫ 3 3 3 Ϫ3 6y 2x 6 ϭ ϩ 6 6 6 1 yϭ xϪ2 3 Ϫ4 Ϫ5 1 yϭ xϩ1 3 Figure 3-4 Because the lines have the same slope but different y-intercepts, they are parallel (Figure 3-4). Two parallel lines do not intersect, which implies that the system has no solution. The system is inconsistent. Skill Practice 4. Solve the system by graphing. 21y Ϫ x2 ϭ 0 Ϫx ϩ y ϭ Ϫ3 Example 5 Solving a System of Linear Equations by Graphing Solve the system by graphing. x ϩ 4y ϭ 8 1 yϭϪ xϩ2 4 Solution: Write the first equation in slope-intercept form. The second equation is already in slope-intercept form. Skill Practice Answers First equation: Second equation: x ϩ 4y ϭ 8 1 yϭϪ xϩ2 4 4. No solution; inconsistent system y 5 4 3 4y ϭ Ϫx ϩ 8 2 1 Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 Ϫ1 Ϫ2 Ϫ3 Ϫ4 Ϫ5 1 2 3 4 5 x 4y Ϫx 8 ϭ ϩ 4 4 4 1 yϭϪ xϩ2 4
7. 7. miL2872X_ch03_177-254 09:22:2006 02:15 PM IA Page 183 Section 3.1 CONFIRMING PAGES 183 Solving Systems of Linear Equations by Graphing y Notice that the slope-intercept forms of the 5 two lines are identical. Therefore, the equa4 tions represent the same line (Figure 3-5). 3 2 The system is dependent, and the solution to 1 y ϭ Ϫ1 x ϩ 2 4 the system of equations is the set of all points x Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 1 2 3 4 5 on the line. Ϫ1 Ϫ2 Because not all the ordered pairs in the Ϫ3 solution set can be listed, we can write the Ϫ4 solution in set-builder notation. Furthermore, Ϫ5 the equations x ϩ 4y ϭ 8 and y ϭ Ϫ1x ϩ 2 Figure 3-5 4 represent the same line. Therefore, the solution set may be written as 5 1x, y2 0 y ϭ Ϫ1x ϩ 26 or 5 1x, y2 0 x ϩ 4y ϭ 86. 4 Skill Practice 5. Solve the system by graphing. 1 yϭ xϩ1 2 x Ϫ 2y ϭ Ϫ2 Calculator Connections The solution to a system of equations can be found by using either a Trace feature or an Intersect feature on a graphing calculator to find the point of intersection between two curves. For example, consider the system Ϫ2x ϩ y ϭ 6 5x ϩ y ϭ Ϫ1 First graph the equations together on the same viewing window. Recall that to enter the equations into the calculator, the equations must be written with the y-variable isolated. That is, be sure to solve for y first. Isolate y. Ϫ2x ϩ y ϭ 6 5x ϩ y ϭ Ϫ1 y ϭ 2x ϩ 6 y ϭ Ϫ5x Ϫ 1 Skill Practice Answers 1 5. { 1x, y2 0 y ϭ x ϩ 1}; infinitely many 2 solutions; dependent system y By inspection of the graph, it appears that the solution is 1Ϫ1, 42. The Trace option on the calculator may come close to 1Ϫ1, 42 but may not show the exact solution (Figure 3-6). However, an Intersect feature on a graphing calculator may provide the exact solution (Figure 3-7). See your user’s manual for further details. 5 4 3 2 1 Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 Ϫ1 Ϫ2 Ϫ3 Ϫ4 Ϫ5 1 2 3 4 5 x
8. 8. miL2872X_ch03_177-254 09:22:2006 184 02:15 PM IA Page 184 CONFIRMING PAGES Chapter 3 Systems of Linear Equations Using Trace Using Intersect Figure 3-6 Section 3.1 Figure 3-7 Practice Exercises Boost your GRADE at mathzone.com! • Practice Problems • Self-Tests • NetTutor • e-Professors • Videos Study Skills Exercises 1. Before you proceed further in Chapter 3, make your test corrections for the Chapter 2 test. See Exercise 1 of Section 2.1 for instructions. 2. Define the key terms. a. System of linear equations b. Solution to a system of linear equations c. Consistent system d. Inconsistent system e. Dependent system f. Independent system Concept 1: Solutions to Systems of Linear Equations For Exercises 3–8, determine which points are solutions to the given system. 3. y ϭ 8x Ϫ 5 y ϭ 4x ϩ 3 1 4. y ϭ Ϫ x Ϫ 5 2 1Ϫ1, 132, 1Ϫ1, 12, (2, 11) yϭ 3 x Ϫ 10 4 9 14, Ϫ72, 10, Ϫ102, a3, Ϫ b 2 6. x ϩ 2y ϭ 4 1 yϭϪ xϩ2 2 1 1Ϫ2, 32, 14, 02, a3, b 2 7. xϪ yϭ6 4x ϩ 3y ϭ Ϫ4 14, Ϫ22, 16, 02, 12, 42 5. 2x Ϫ 7y ϭ Ϫ30 y ϭ 3x ϩ 7 3 10, Ϫ302 , a , 5b, 1Ϫ1, 42 2 8. x Ϫ 3y ϭ 3 2x Ϫ 9y ϭ 1 10, 12, 14, Ϫ12, 19, 22
9. 9. miL2872X_ch03_177-254 09:22:2006 02:15 PM IA Page 185 Section 3.1 CONFIRMING PAGES 185 Solving Systems of Linear Equations by Graphing Concept 2: Dependent and Inconsistent Systems of Linear Equations For Exercises 9–14, the graph of a system of linear equations is given. a. Identify whether the system is consistent or inconsistent. b. Identify whether the system is dependent or independent. c. Identify the number of solutions to the system. 9. yϭxϩ3 10. 5x Ϫ 3y ϭ 6 3x ϩ y ϭ Ϫ1 11. 2x ϭ y ϩ 4 3y ϭ 2x ϩ 3 y Ϫ4x ϩ 2y ϭ 2 y y 5 4 3 5 4 3 5 4 3 2 1 2 1 2 1 Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 Ϫ1 Ϫ2 1 2 3 4 5 x Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 Ϫ1 Ϫ2 1 2 3 4 5 x Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 Ϫ1 Ϫ2 Ϫ3 12. Ϫ3 Ϫ4 Ϫ5 2 3 4 5 x Ϫ3 Ϫ4 Ϫ5 1 Ϫ4 Ϫ5 y ϭ Ϫ2x Ϫ 3 13. Ϫ4x Ϫ 2y ϭ 0 1 yϭ xϩ2 3 Ϫx ϩ 3y ϭ 6 y 2 yϭϪ xϪ1 3 14. Ϫ4x Ϫ 6y ϭ 6 y y 5 4 3 5 4 3 5 4 3 2 1 2 1 2 1 Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 Ϫ1 Ϫ2 1 2 3 4 5 x Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 Ϫ1 Ϫ2 1 2 3 4 5 x Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 Ϫ1 Ϫ2 Ϫ3 Ϫ3 Ϫ4 Ϫ5 2 3 4 5 1 2 3 4 5 x Ϫ3 Ϫ4 Ϫ5 1 Ϫ4 Ϫ5 Concept 3: Solving Systems of Linear Equations by Graphing For Exercises 15–32, solve the systems of equations by graphing. 15. 2x ϩ y ϭ 4 16. 4x Ϫ 3y ϭ 12 x ϩ 2y ϭ Ϫ1 17. y ϭ Ϫ2x ϩ 3 3x ϩ 4y ϭ Ϫ16 y yϭ 5x Ϫ 4 y y 5 4 3 5 4 3 5 4 3 2 1 2 1 2 1 Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 Ϫ1 Ϫ2 1 2 3 4 5 x Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 Ϫ1 Ϫ2 1 2 3 4 5 x Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 Ϫ1 Ϫ2 Ϫ3 Ϫ3 Ϫ3 Ϫ4 Ϫ5 Ϫ4 Ϫ5 Ϫ4 Ϫ5 x
10. 10. miL2872X_ch03_177-254 09:22:2006 02:15 PM 186 IA Page 186 CONFIRMING PAGES Chapter 3 Systems of Linear Equations 1 xϪ5 3 1 20. y ϭ x ϩ 2 2 2 yϭϪ xϪ2 3 18. y ϭ 2x ϩ 5 5 yϭ xϪ2 2 19. y ϭ y ϭ Ϫx ϩ 2 y y y 5 4 3 5 4 3 5 4 3 2 1 2 1 2 1 Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 Ϫ1 Ϫ2 1 2 3 4 5 x Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 Ϫ1 Ϫ2 1 2 3 4 5 x Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 Ϫ1 Ϫ2 Ϫ3 Ϫ3 Ϫ4 Ϫ5 2 3 4 5 1 2 3 4 5 1 2 3 4 5 x Ϫ3 Ϫ4 Ϫ5 1 Ϫ4 Ϫ5 21. x ϭ 4 22. 3x ϩ 2y ϭ 6 y ϭ 2x Ϫ 3 23. y ϭ Ϫ2x ϩ 3 y ϭ Ϫ3 Ϫ2x ϭ y ϩ 1 y y y 5 4 3 5 4 3 5 4 3 2 1 2 1 2 1 Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 Ϫ1 Ϫ2 1 2 3 4 5 x Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 Ϫ1 Ϫ2 1 2 3 4 5 x Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 Ϫ1 Ϫ2 Ϫ3 Ϫ3 Ϫ3 Ϫ4 Ϫ5 Ϫ4 Ϫ5 Ϫ4 Ϫ5 1 24. y ϭ x Ϫ 2 3 2 25. y ϭ x Ϫ 1 3 x x ϭ 3y Ϫ 9 26. 4x ϭ 16 Ϫ 8y 1 yϭϪ xϩ2 2 2x ϭ 3y ϩ 3 y y y 5 4 3 5 4 3 5 4 3 2 1 2 1 2 1 Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 Ϫ1 Ϫ2 1 2 3 4 5 x Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 Ϫ1 Ϫ2 1 2 3 4 5 x Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 Ϫ1 Ϫ2 Ϫ3 Ϫ3 Ϫ3 Ϫ4 Ϫ5 Ϫ4 Ϫ5 Ϫ4 Ϫ5 x
11. 11. miL2872X_ch03_177-254 09:22:2006 02:15 PM Section 3.1 27. 2x ϭ 4 IA Page 187 CONFIRMING PAGES 28. y ϩ 7 ϭ 6 29. Ϫx ϩ 3y ϭ 6 6y ϭ 2x ϩ 12 Ϫ5 ϭ 2x 1 y ϭ Ϫ1 2 187 Solving Systems of Linear Equations by Graphing y y y 5 4 3 5 4 3 5 4 3 2 1 2 1 2 1 Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 Ϫ1 Ϫ2 1 2 3 4 5 x Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 Ϫ1 Ϫ2 1 2 3 4 5 x Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 Ϫ1 Ϫ2 Ϫ3 Ϫ3 Ϫ4 Ϫ5 3 4 5 3 4 5 x Ϫ4 Ϫ5 31. 2x Ϫ y ϭ 4 2 Ϫ3 Ϫ4 Ϫ5 1 32. x ϭ 4y ϩ 4 30. 3x ϭ 2y Ϫ 4 Ϫ4y ϭ Ϫ6x Ϫ 8 4x ϩ 2 ϭ 2y y Ϫ2x ϩ 8y ϭ Ϫ16 y y 5 4 3 5 4 3 5 4 3 2 1 2 1 2 1 Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 Ϫ1 Ϫ2 1 2 3 4 5 x Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 Ϫ1 Ϫ2 1 2 3 4 5 x Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 Ϫ1 Ϫ2 Ϫ3 Ϫ3 Ϫ4 Ϫ5 2 x Ϫ3 Ϫ4 Ϫ5 1 Ϫ4 Ϫ5 For Exercises 33–36, identify each statement as true or false. 33. A consistent system is a system that always has a unique solution. 34. A dependent system is a system that has no solution. 35. If two lines coincide, the system is dependent. 36. If two lines are parallel, the system is independent. Graphing Calculator Exercises For Exercises 37–42, use a graphing calculator to graph each linear equation on the same viewing window. Use a Trace or Intersect feature to find the point(s) of intersection. 37. y ϭ 2x Ϫ 3 y ϭ Ϫ4x ϩ 9 1 38. y ϭ Ϫ x ϩ 2 2 1 yϭ xϪ3 3
12. 12. miL2872X_ch03_177-254 09:22:2006 188 39. 02:16 PM IA Page 188 CONFIRMING PAGES Chapter 3 Systems of Linear Equations xϩyϭ4 40. Ϫ2x ϩ y ϭ Ϫ5 Ϫ3x ϩ 2y ϭ 6 41. Ϫx ϩ 3y ϭ Ϫ6 42. x ϩ 4y ϭ 8 6y ϭ 2x ϩ 6 Section 3.2 Concepts 1. The Substitution Method 2. Solving Inconsistent Systems and Dependent Systems x Ϫ 2y ϭ Ϫ2 1 yϭϪ xϩ2 4 Solving Systems of Equations by Using the Substitution Method 1. The Substitution Method Graphing a system of equations is one method to find the solution of the system. In this section and Section 3.3, we will present two algebraic methods to solve a system of equations. The first is called the substitution method. This technique is particularly important because it can be used to solve more advanced problems including nonlinear systems of equations. The first step in the substitution process is to isolate one of the variables from one of the equations. Consider the system x ϩ y ϭ 16 xϪyϭ4 Solving the first equation for x yields x ϭ 16 Ϫ y. Then, because x is equal to 16 Ϫ y, the expression 16 Ϫ y can replace x in the second equation. This leaves the second equation in terms of y only. Solve for x. x ϩ y ϭ 16 Second equation: 116 Ϫ y2 Ϫ y ϭ 4 x ϭ 16 Ϫ y v First equation: 16 Ϫ 2y ϭ 4 Substitute x ϭ 16 Ϫ y. Solve for y. Ϫ2y ϭ Ϫ12 yϭ6 x ϭ 16 Ϫ y x ϭ 16 Ϫ 162 x ϭ 10 To find x, substitute y ϭ 6 back into the equation x ϭ 16 Ϫ y. The solution is (10, 6).
13. 13. miL2872X_ch03_177-254 09:22:2006 02:16 PM Section 3.2 IA Page 189 Solving Systems of Equations by Using the Substitution Method Solving a System of Equations by the Substitution Method 1. Isolate one of the variables from one equation. 2. Substitute the quantity found in step 1 into the other equation. 3. Solve the resulting equation. 4. Substitute the value found in step 3 back into the equation in step 1 to find the value of the remaining variable. 5. Check the solution in both equations, and write the answer as an ordered pair. Example 1 Using the Substitution Method to Solve a Linear Equation Solve the system by using the substitution method. Ϫ3x ϩ 4y ϭ 9 1 xϭϪ yϩ2 3 Solution: Ϫ3x ϩ 4y ϭ 9 ¶ 1 xϭϪ yϩ2 3 1 Ϫ3aϪ y ϩ 2b ϩ 4y ϭ 9 3 y Ϫ 6 ϩ 4y ϭ 9 Step 1: In the second equation, x is already isolated. 1 Step 2: Substitute the quantity Ϫ y ϩ 2 for x 3 in the other equation. Step 3: Solve for y. 5y ϭ 15 yϭ3 Now use the known value of y to solve for the remaining variable x. 1 xϭϪ yϩ2 3 1 x ϭ Ϫ 132 ϩ 2 3 x ϭ Ϫ1 ϩ 2 Step 4: Substitute y ϭ 3 into the equation 1 x ϭ Ϫ y ϩ 2. 3 xϭ1 Step 5: Check the ordered pair (1, 3) in each original equation. Ϫ3x ϩ 4y ϭ 9 Ϫ3112 ϩ 4132 ՘ 9 Ϫ3 ϩ 12 ϭ 9 ✔ True The solution is (1, 3). CONFIRMING PAGES 1 xϭϪ yϩ2 3 1 1 ՘ Ϫ 132 ϩ 2 3 1 ϭ Ϫ1 ϩ 2 ✔ True 189
14. 14. miL2872X_ch03_177-254 09:22:2006 190 02:16 PM IA Page 190 CONFIRMING PAGES Chapter 3 Systems of Linear Equations Skill Practice 1. Solve by using the substitution method. x ϭ 2y ϩ 3 4x Ϫ 2y ϭ 0 Using the Substitution Method to Solve a Linear System Example 2 Solve the system by using the substitution method. 3x Ϫ 2y ϭ Ϫ7 6x ϩ y ϭ 6 Solution: The y variable in the second equation is the easiest variable to isolate because its coefficient is 1. 3x Ϫ 2y ϭ Ϫ7 y ϭ Ϫ6x ϩ 6 ¶ 6x ϩ y ϭ 6 3x Ϫ 21Ϫ6x ϩ 62 ϭ Ϫ7 Step 1: Solve the second equation for y. Step 2: Substitute the quantity Ϫ6x ϩ 6 for y in the other equation. 3x ϩ 12x Ϫ 12 ϭ Ϫ7 15x Ϫ 12 ϭ Ϫ7 15x ϭ 5 Step 3: Solve for x. 5 15x ϭ 15 15 xϭ 1 3 y ϭ Ϫ6x ϩ 6 Avoiding Mistakes: Do not substitute y ϭ Ϫ6x ϩ 6 into the same equation from which it came. This mistake will result in an identity: 6x ϩ y ϭ 6 6x ϩ 1Ϫ6x ϩ 62 ϭ 6 6x Ϫ 6x ϩ 6 ϭ 6 6ϭ6 1 y ϭ Ϫ6 a b ϩ 6 3 y ϭ Ϫ2 ϩ 6 yϭ4 3x Ϫ 2y ϭ Ϫ7 6x ϩ y ϭ 6 1 3 a b Ϫ 2142 ՘ Ϫ7 3 1 6a b ϩ 4 ՘ 6 3 1 Ϫ 8 ϭ Ϫ7 ✔ The solution is 1 1, 42. 3 Skill Practice Answers 1. (Ϫ1, Ϫ2) Step 4: Substitute x ϭ 1 into 3 the equation y ϭ Ϫ6x ϩ 6. 2ϩ4ϭ6✔ Step 5: Check the ordered pair 1 1, 42 in each original 3 equation.
15. 15. miL2872X_ch03_177-254 09:22:2006 02:16 PM Section 3.2 IA Page 191 CONFIRMING PAGES 191 Solving Systems of Equations by Using the Substitution Method Skill Practice 2. Solve by the substitution method. 3x ϩ y ϭ 8 x Ϫ 2y ϭ 12 2. Solving Inconsistent Systems and Dependent Systems Example 3 Using the Substitution Method to Solve a Linear System Solve the system by using the substitution method. x ϭ 2y Ϫ 4 Ϫ2x ϩ 4y ϭ 6 Solution: x ϭ 2y Ϫ 4 Step 1: The x variable is already isolated. Ϫ2x ϩ 4y ϭ 6 Ϫ212y Ϫ 42 ϩ 4y ϭ 6 Step 2: Substitute the quantity x ϭ 2y Ϫ 4 into the other equation. Ϫ4y ϩ 8 ϩ 4y ϭ 6 Step 3: Solve for y. 8ϭ6 There is no solution. The system is inconsistent. The equation reduces to a contradiction, indicating that the system has no solution. The lines never intersect and must be parallel. The system is inconsistent. TIP: The answer to Example 3 can be verified by writing each equation in slope-intercept form and graphing the equations. Equation 1 x ϭ 2y Ϫ 4 Equation 2 y Ϫ2x ϩ 4y ϭ 6 2y ϭ x ϩ 4 4y ϭ 2x ϩ 6 2y x 4 ϭ ϩ 2 2 2 4y 2x 6 ϭ ϩ 4 4 4 1 yϭ xϩ2 2 1 3 yϭ xϩ 2 2 5 x ϭ 2y Ϫ 4 4 3 Ϫ2x ϩ 4y ϭ 6 2 1 x 1 2 3 4 5 Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 Ϫ1 Ϫ2 Ϫ3 Ϫ4 Notice that the equations have the same slope, but different y-intercepts; therefore, the lines must be parallel. There is no solution to this system of equations. Skill Practice 3. Solve by the substitution method. 8x Ϫ 16y ϭ 3 1 yϭ xϩ1 2 Skill Practice Answers 2. (4, Ϫ4) 3. No solution; Inconsistent system
16. 16. miL2872X_ch03_177-254 09:22:2006 192 02:16 PM IA Page 192 CONFIRMING PAGES Chapter 3 Systems of Linear Equations Solving a Dependent System Example 4 Solve by using the substitution method. 4x Ϫ 2y ϭ Ϫ6 y Ϫ 3 ϭ 2x Solution: 4x Ϫ 2y ϭ Ϫ6 y ϭ 2x ϩ 3 Step 1: Solve for one of the variables. v y Ϫ 3 ϭ 2x 4x Ϫ 212x ϩ 32 ϭ Ϫ6 Step 2: Substitute the quantity 2x ϩ 3 for y in the other equation. 4x Ϫ 4x Ϫ 6 ϭ Ϫ6 Step 3: Solve for x. Ϫ6 ϭ Ϫ6 The system reduces to the identity Ϫ6 ϭ Ϫ6. Therefore, the original two equations are equivalent, and the system is dependent. The solution consists of all points on the common line. Because the equations 4x Ϫ 2y ϭ Ϫ6 and y Ϫ 3 ϭ 2x represent the same line, the solution may be written as 5 1x, y2 0 4x Ϫ 2y ϭ Ϫ66 or 51x, y2 0 y Ϫ 3 ϭ 2x6 Skill Practice 4. Solve the system by using substitution. 3x ϩ 6y ϭ 12 2y ϭ Ϫx ϩ 4 TIP: We can confirm the results of Example 4 by writing each equation in slope-intercept form. The slope-intercept forms are identical, indicating that the lines are the same. slope-intercept form Skill Practice Answers 4. Infinitely many solutions; 51x, y2 0 3x ϩ 6y ϭ 126; Dependent system Section 3.2 Boost your GRADE at mathzone.com! 4x Ϫ 2y ϭ Ϫ6 Ϫ2y ϭ Ϫ4x Ϫ 6 y Ϫ 3 ϭ 2x y ϭ 2x ϩ 3 y ϭ 2x ϩ 3 Practice Exercises • Practice Problems • Self-Tests • NetTutor • e-Professors • Videos Study Skills Exercise 1. Check your progress by answering these questions. Yes ____ No ____ Did you have sufficient time to study for the test on Chapter 2? If not, what could you have done to create more time for studying?
17. 17. miL2872X_ch03_177-254 9/23/06 02:14 PM IA Page 193 CONFIRMING PAGES 193 Section 3.2 Solving Systems of Equations by Using the Substitution Method Yes ____ No ____ Did you work all of the assigned homework problems in Chapter 2? Yes ____ No ____ If you encountered difficulty, did you see your instructor or tutor for help? Yes ____ No ____ Have you taken advantage of the textbook supplements such as the Student Solutions Manual and MathZone? Review Exercises For Exercises 2–5, using the slope-intercept form of the lines, a. determine whether the system is consistent or inconsistent and b. determine whether the system is dependent or independent. 2. y ϭ 8x Ϫ 1 2x Ϫ 16y ϭ 3 3. 4x ϩ 6y ϭ 1 4. 2x Ϫ 4y ϭ 0 5 2 x Ϫ 2y ϭ 9 10x ϩ 15y ϭ 5. 6x ϩ 3y ϭ 8 8x ϩ 4y ϭ Ϫ1 For Exercises 6–7, solve the system by graphing. 6. xϪyϭ4 7. y ϭ 2x ϩ 3 y 5 4 3 2 3x ϩ 4y ϭ 12 y 5 4 3 2 6x ϩ 3y ϭ 9 1 Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 Ϫ1 Ϫ2 1 1 2 3 4 5 x Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 Ϫ1 Ϫ2 Ϫ3 2 3 4 Ϫ3 Ϫ4 Ϫ5 1 Ϫ4 Ϫ5 Concept 1: The Substitution Method For Exercises 8–17, solve by using the substitution method. 8. 4x ϩ 12y ϭ 4 9. y ϭ 5x ϩ 11 11. Ϫ3x ϩ 8y ϭ Ϫ1 12. 12x Ϫ 2y ϭ x Ϫ 3y ϭ Ϫ4 2x ϩ 3y ϭ Ϫ5 x ϭ 10y ϩ 34 10. 2x Ϫ 3y ϭ Ϫ8 Ϫ7x ϩ y ϭ Ϫ31 15. 0 13. 3x ϩ 12y ϭ 24 Ϫ7x ϩ y ϭ Ϫ1 4x Ϫ y ϭ 11 14. y ϭ Ϫ3x Ϫ 1 x Ϫ 5y ϭ 17 xϪ yϭ8 16. 5x Ϫ 2y ϭ 10 3x ϩ 2y ϭ 9 yϭxϪ1 17. 2x Ϫ y ϭ Ϫ1 y ϭ Ϫ2x 18. Describe the process of solving a system of linear equations by using substitution. Concept 2: Solving Inconsistent Systems and Dependent Systems For Exercises 19–26, solve the systems. 19. 2x Ϫ 6y ϭ Ϫ2 x ϭ 3y Ϫ 1 20. Ϫ2x ϩ 4y ϭ 22 x ϭ 2y Ϫ 11 21. yϭ 1 xϩ3 7 x Ϫ 7y ϭ Ϫ4 5 x
18. 18. miL2872X_ch03_177-254 09:22:2006 194 22. 02:17 PM CONFIRMING PAGES Chapter 3 Systems of Linear Equations 3 1 xϭϪ yϩ 2 2 23. 5x Ϫ y ϭ 10 3x Ϫ y ϭ 7 24. x ϩ 4y ϭ 8 2y ϭ 10x Ϫ 5 4x ϩ 6y ϭ 7 25. IA Page 194 26. Ϫ14 ϩ 6x ϭ 2y xϭ 3x ϭ 3 Ϫ 12y 4y ϩ 1 Ϫ12y ϭ Ϫ3x ϩ 3 27. When using the substitution method, explain how to determine whether a system of linear equations is dependent. 28. When using the substitution method, explain how to determine whether a system of linear equations is inconsistent. Mixed Exercises For Exercises 29–50, solve the system by using the substitution method. 29. x ϭ 1.3y ϩ 1.5 y ϭ 1.2x Ϫ 4.6 1 5 32. x ϭ y Ϫ 6 3 1.1x ϭ Ϫy ϩ 9.6 200y ϭ 150x yϪ4ϭ1 41. y ϭ 200x Ϫ 320 y ϭ Ϫ150x ϩ 1080 44. y ϭ 6.8x ϩ 2.3 y ϭ Ϫ4.1x ϩ 56.8 47. 21x ϩ 2y2 ϭ 12 Ϫ6x ϭ 5y Ϫ 8 34. 1 1 1 Ϫ xϩ yϭ 4 8 4 8x Ϫ y ϭ 8 1 1 1 xϪ yϭ 3 24 2 36. Ϫx ϩ 4y ϭ Ϫ4 yϭxϩ3 38. 1 17 xϭ yϩ 4 4 33. Ϫ2x ϩ y ϭ 4 1 21 yϭ xϩ 5 5 35. 3x ϩ 2y ϭ 6 1 2 31. y ϭ x Ϫ 3 3 30. y ϭ 0.8x Ϫ 1.8 37. Ϫ300x Ϫ 125y ϭ 1350 yϭxϪ1 39. yϩ2ϭ8 2x Ϫ y ϭ 6 40. 1 1 1 xϪ yϭ 6 12 2 1 1 1 xϪ yϭ 16 4 2 42. y ϭ Ϫ54x ϩ 300 43. y ϭ Ϫ2.7x Ϫ 5.1 y ϭ 20x Ϫ 70 y ϭ 3.1x Ϫ 63.1 45. 4x ϩ 4y ϭ 5 x Ϫ 4y ϭ Ϫ 48. 5x Ϫ 2y ϭ Ϫ25 10x ϭ 31y Ϫ 102 x Ϫ 4y ϭ 8 46. Ϫ2x ϩ y ϭ Ϫ6 6x Ϫ 13y ϭ Ϫ12 5 2 49. 513x Ϫ y2 ϭ 10 4y ϭ 7x Ϫ 3 50. 2x ϭ Ϫ31y ϩ 32 3x Ϫ 4y ϭ Ϫ22
20. 20. miL2872X_ch03_177-254 196 09:22:2006 02:17 PM IA Page 196 CONFIRMING PAGES Chapter 3 Systems of Linear Equations The steps to solve a system of linear equations in two variables by the addition method is outlined in the following box. Solving a System of Equations by the Addition Method 1. Write both equations in standard form: Ax ϩ By ϭ C 2. Clear fractions or decimals (optional). 3. Multiply one or both equations by nonzero constants to create opposite coefficients for one of the variables. 4. Add the equations from step 3 to eliminate one variable. 5. Solve for the remaining variable. 6. Substitute the known value found in step 5 into one of the original equations to solve for the other variable. 7. Check the ordered pair in both equations. Solving a System by the Addition Method Example 2 Solve the system by using the addition method. 4x ϩ 5y ϭ 2 3x ϭ 1 Ϫ 4y Solution: 4x ϩ 5y ϭ 2 4x ϩ 5y ϭ 2 3x ϭ 1 Ϫ 4y 3x ϩ 4y ϭ 1 Step 1: Write both equations in standard form. There are no fractions or decimals. We may choose to eliminate either variable. To eliminate x, change the coefficients to 12 and Ϫ12. 4x ϩ 5y ϭ 2 Multiply by 3. 12x ϩ 15y ϭ 6 Multiply by Ϫ4. 3x ϩ 4y ϭ 1 Ϫ12x Ϫ 16y ϭ Ϫ4 12x ϩ 15y ϭ 6 Ϫ12x Ϫ 16y ϭ Ϫ4 Ϫy ϭ 2 Step 3: Multiply the first equation by 3. Multiply the second equation by Ϫ4. Step 4: Add the equations. Step 5: Solve for y. y ϭ Ϫ2 4x ϩ 5y ϭ 2 4x ϩ 51Ϫ22 ϭ 2 4x Ϫ 10 ϭ 2 4x ϭ 12 Step 6: Substitute y ϭ Ϫ2 back into one of the original equations and solve for x. xϭ3 The solution is (3, Ϫ2). Step 7: The ordered pair (3, Ϫ2) checks in both original equations.
21. 21. miL2872X_ch03_177-254 09:22:2006 02:17 PM Section 3.3 Page 197 IA CONFIRMING PAGES 197 Solving Systems of Equations by Using the Addition Method Skill Practice 2. Solve by the addition method. 2y ϭ 5x Ϫ 4 3x Ϫ 4y ϭ 1 TIP: To eliminate the x variable in Example 2, both equations were multiplied by appropriate constants to create 12x and Ϫ12x. We chose 12 because it is the least common multiple of 4 and 3. We could have solved the system by eliminating the y-variable. To eliminate y, we would multiply the top equation by 4 and the bottom equation by Ϫ5. This would make the coefficients of the y-variable 20 and Ϫ20, respectively. 4x ϩ 5y ϭ 2 3x ϩ 4y ϭ 1 Multiply by 4. Multiply by Ϫ5. Example 3 16x ϩ 20y ϭ 8 Ϫ15x Ϫ 20y ϭ Ϫ5 Solving a System of Equations by the Addition Method Solve the system by using the addition method. x Ϫ 2y ϭ 6 ϩ y 0.05y ϭ 0.02x Ϫ 0.10 Solution: x Ϫ 2y ϭ 6 ϩ y x Ϫ 3y ϭ 6 0.05y ϭ 0.02x Ϫ 0.10 x Ϫ0.02x ϩ 0.05y ϭ Ϫ0.10 Step 1: Write both equations in standard form. Ϫ 3y ϭ 6 Ϫ0.02x ϩ 0.05y ϭ Ϫ0.10 x Ϫ 3y ϭ 6 Multiply by 100. Multiply by 2. Ϫ2x ϩ 5y ϭ Ϫ10 Ϫ2x ϩ 5y ϭ Ϫ10 2x Ϫ 6y ϭ 12 Ϫ2x ϩ 5y ϭ Ϫ10 Ϫy ϭ 2 y ϭ Ϫ2 x Ϫ 2y ϭ 6 ϩ y x Ϫ 21Ϫ22 ϭ 6 ϩ 1Ϫ22 xϩ4ϭ4 xϭ0 Step 2: Clear decimals. Step 3: Create opposite coefficients. Step 4: Add the equations. Step 5: Solve for y. Step 6: To solve for x, substitute y ϭ Ϫ2 into one of the original equations. Skill Practice Answers 1 2. a1, b 2
22. 22. miL2872X_ch03_177-254 09:22:2006 198 02:17 PM IA Page 198 CONFIRMING PAGES Chapter 3 Systems of Linear Equations Step 7: Check the ordered pair (0, Ϫ2) in each original equation. x Ϫ 2y ϭ 6 ϩ y 0.05y ϭ 0.02x Ϫ 0.10 102 Ϫ 21Ϫ22 ՘ 6 ϩ 1Ϫ22 4ϭ4✔ 0.051Ϫ22 ՘ 0.02102 Ϫ 0.10 Ϫ0.10 ϭ Ϫ0.10 ✔ The solution is (0, Ϫ2). Skill Practice 3. Solve by the addition method. 0.2x ϩ 0.3y ϭ 1.5 5x ϩ 3y ϭ 20 Ϫ y 2. Solving Inconsistent Systems and Dependent Systems Example 4 Solving a System of Equations by the Addition Method Solve the system by using the addition method. 1 1 xϪ yϭ1 5 2 Ϫ4x ϩ 10y ϭ Ϫ20 Solution: 1 1 x Ϫ y ϭ1 5 2 Step 1: Equations are in standard form. Ϫ4x ϩ 10y ϭ Ϫ20 1 1 10 a x Ϫ yb ϭ 10 ؒ 1 5 2 2x Ϫ 5y ϭ 10 Step 2: Clear fractions. 4x Ϫ 10y ϭ 20 Step 3: Multiply the first equation by 2. Ϫ4x ϩ 10y ϭ Ϫ20 2x Ϫ 5y ϭ 10 Ϫ4x ϩ 10y ϭ Ϫ20 Multiply by 2. Ϫ4x ϩ 10y ϭ Ϫ20 0ϭ0 Step 4: Add the equations. Notice that both variables were eliminated. The system of equations is reduced to the identity 0 ϭ 0. Therefore, the two original equations are equivalent and the system is dependent. The solution set consists of an infinite number of ordered pairs (x, y) that fall on the common line of intersection Ϫ4x ϩ 10y ϭ Ϫ20, or equivalently 1x Ϫ 1y ϭ 1. The solution set can be written in set notation as 5 2 5 1x, y2 0 Ϫ4x ϩ 10y ϭ Ϫ206 Skill Practice Answers 3. (0, 5) or 1 1 e 1x, y2 ` x Ϫ y ϭ 1 f 5 2
23. 23. miL2872X_ch03_177-254 09:22:2006 02:17 PM Section 3.3 IA Page 199 CONFIRMING PAGES 199 Solving Systems of Equations by Using the Addition Method Skill Practice 4. Solve by the addition method. 3x ϩ y ϭ 4 1 4 xϭϪ yϩ 3 3 Example 5 Solving an Inconsistent System 2y ϭ Ϫ3x ϩ 4 Solve the system by using the addition method. 120x ϩ 80y ϭ 40 Solution: 2y ϭ Ϫ3x ϩ 4 Standard form 120x ϩ 80y ϭ 40 3x ϩ 2y ϭ 4 Step 1: Write the equations in standard form. 120x ϩ 80y ϭ 40 Step 2: There are no decimals or fractions. 3x ϩ 2y ϭ 4 Multiply by Ϫ40. Ϫ120x Ϫ 80y ϭ Ϫ160 120x ϩ 80y ϭ 40 120x ϩ 80y ϭ 40 0 ϭ Ϫ120 Step 3: Multiply the top equation by Ϫ40. Step 4: Add the equations. The equations reduce to a contradiction, indicating that the system has no solution. The system is inconsistent. The two equations represent parallel lines, as shown in Figure 3-8. There is no solution. y 120x ϩ 80y ϩ 40 5 4 3 2 1 Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 Ϫ1 Ϫ2 2y ϭ Ϫ3x ϩ 4 1 2 3 4 5 x Ϫ3 Ϫ4 Ϫ5 Figure 3-8 Skill Practice 5. Solve by the addition method. 18 ϩ 10x ϭ 6y 5x Ϫ 3y ϭ 9 Skill Practice Answers 4. Infinitely many solutions; {(x, y) 0 3x ϩ y ϭ 4}; Dependent system 5. No solution; Inconsistent system
24. 24. miL2872X_ch03_177-254 09:22:2006 02:17 PM 200 IA Page 200 CONFIRMING PAGES Chapter 3 Systems of Linear Equations Section 3.3 Practice Exercises Boost your GRADE at mathzone.com! • Practice Problems • Self-Tests • NetTutor • e-Professors • Videos Study Skills Exercise 1. Instructors differ in what they emphasize on tests. For example, test material may come from the textbook, notes, handouts, or homework. What does your instructor emphasize? Review Exercises For Exercises 2–4, use the slope-intercept form of the lines to determine the number of solutions for the system of equations. 1 2. y ϭ x Ϫ 4 2 3. y ϭ 2.32x Ϫ 8.1 y ϭ 1.46x Ϫ 8.1 1 yϭ xϩ1 2 4. 4x ϭ y ϩ 7 Ϫ2y ϭ Ϫ8x ϩ 14 Concept 1: The Addition Method For Exercises 5–14, solve the system by the addition method. 5. 3x Ϫ y ϭ Ϫ1 6. 5x Ϫ 2y ϭ 15 Ϫ3x ϩ 4y ϭ Ϫ14 8. 2x Ϫ 5y ϭ 7 3x Ϫ 10y ϭ 13 11. 3x ϭ 10y ϩ 13 7y ϭ 4x Ϫ 11 3x ϩ 2y ϭ Ϫ7 2x ϩ 3y ϭ 3 7. Ϫ10x ϩ 2y ϭ Ϫ32 3x ϩ 7y ϭ Ϫ20 10. 6x Ϫ 9y ϭ Ϫ15 Ϫ5x ϩ 3y ϭ Ϫ84 9. 5x Ϫ 2y ϭ Ϫ40 12. Ϫ5x ϭ 6y Ϫ 4 13. 1.2x Ϫ 0.6y ϭ 3 5y ϭ 1 Ϫ 3x 0.8x Ϫ 1.4y ϭ 3 14. 1.8x ϩ 0.8y ϭ 1.4 1.2x ϩ 0.6y ϭ 1.2 Concept 2: Solving Inconsistent Systems and Dependent Systems For Exercises 15–22, solve the systems. 15. 3x Ϫ 2y ϭ 1 Ϫ6x ϩ 4y ϭ Ϫ2 18. 2x ϭ 4 Ϫ y Ϫ0.1y ϭ 0.2x Ϫ 0.2 21. 7 1 xϩ yϭ 2 6 x ϩ 2y ϭ 4.5 16. 3x Ϫ y ϭ 4 6x Ϫ 2y ϭ 8 19. 12x Ϫ 4y ϭ 2 0.6x ϭ 0.1 ϩ 0.2y 22. 0.2x Ϫ 0.1y ϭ Ϫ1.2 xϪ 1 yϭ3 2 17. 6y ϭ 14 Ϫ 4x 0.2x ϭ Ϫ0.3y Ϫ 0.7 20. 10x Ϫ 15y ϭ 5 0.3y ϭ 0.2x Ϫ 0.1
25. 25. miL2872X_ch03_177-254 09:22:2006 02:17 PM Section 3.3 IA Page 201 CONFIRMING PAGES 201 Solving Systems of Equations by Using the Addition Method Mixed Exercises 23. Describe a situation in which you would prefer to use the substitution method over the addition method. 24. If you used the addition method to solve the given system, would it be easier to eliminate the x- or yvariable? Explain. 3x Ϫ 5y ϭ 4 7x ϩ 10y ϭ 31 For Exercises 25–50, solve by using either the addition method or the substitution method. 25. 2x Ϫ 4y ϭ 8 26. 8x ϩ 6y ϭ Ϫ8 y ϭ 2x ϩ 1 27. 2x ϩ 5y ϭ 9 x ϭ 6y Ϫ 10 x ϩ 5y ϭ 7 29. 2x Ϫ y ϭ 8 2x ϩ 7y ϭ 8 xϪyϭ4 31. 0.4x Ϫ 0.6y ϭ 0.5 32. 0.3x ϩ 0.6y ϭ 0.7 0.2x Ϫ 0.3y ϭ 0.7 4x Ϫ 7y ϭ Ϫ16 1 30. y ϭ x Ϫ 3 2 0.2x ϩ 0.4y ϭ 0.5 28. 34. 1 1 xϩ yϭ7 3 5 35. 1 2 x Ϫ y ϭ Ϫ4 6 5 37. 21x ϩ 2y2 ϭ 20 Ϫ y 38. Ϫ31x ϩ y2 ϭ 10 Ϫ 4y xϩ 46. 4 yϭ xϩ2 3 2 2 xϪ yϭ0 5 3 3 yϭ x 5 39. Ϫ4y ϭ 10 4x ϩ 3 ϭ 1 5x Ϫ 3y ϭ 18 42. 6x Ϫ 3y ϭ Ϫ3 4x ϩ 5y ϭ Ϫ9 44. 212y ϩ 32 Ϫ 2x ϭ 1 Ϫ x 1 x ϩ y ϭ 17 ϩ y2 5 47. 4x ϩ y ϭ Ϫ2 5x Ϫ y ϭ Ϫ7 1 11 xϩ yϭϪ 4 2 49. 4x ϭ 3y 36. Ϫ3x ϩ 5y ϭ 18 41. 2 12y Ϫ 32 ϭ Ϫ2 3 1 8 1 xϪ yϭϪ 10 2 5 1 1 x Ϫ y ϭ Ϫ2 4 6 1 1 Ϫ xϩ yϭ4 6 5 41x ϩ 2y2 ϭ 50 ϩ 3y 3y ϩ 2 ϭ 1 1 43. 3x Ϫ 2 ϭ 111 ϩ 5y2 3 33. 3 xϭ y 2 Ϫ71x Ϫ y2 ϭ 16 ϩ 3y 40. Ϫ9x ϭ 15 1 1 xϪ yϭ0 3 2 4x ϩ y ϭ Ϫ3 50. 4x Ϫ 2y ϭ 6 1 3 xϭ yϩ 2 2 45. 1 11 1 xϩ yϭ 4 2 4 2 1 7 xϩ yϭ 3 3 3 48. 4y ϭ 8x ϩ 20 8x ϭ 24
26. 26. miL2872X_ch03_177-254 09:22:2006 02:17 PM 202 IA Page 202 CONFIRMING PAGES Chapter 3 Systems of Linear Equations Expanding Your Skills For Exercises 51–54, use the addition method first to solve for x. Then repeat the addition method again, using the original system of equations, this time solving for y. 51. 6x Ϫ 2y ϭ 5 5x ϩ 3y ϭ Ϫ2 Section 3.4 Concepts 1. Applications Involving Cost 2. Applications Involving Mixtures 3. Applications Involving Principal and Interest 4. Applications Involving Distance, Rate, and Time 5. Applications Involving Geometry 5x Ϫ 4y ϭ 7 52. 4x ϩ 7y ϭ 8 53. Ϫ2x ϩ 6y ϭ 5 54. Ϫ5x Ϫ 2y ϭ 3 2x Ϫ 4y ϭ 9 Ϫ5x ϩ 3y ϭ 1 Applications of Systems of Linear Equations in Two Variables 1. Applications Involving Cost In Chapter 1 we solved numerous application problems using equations that contained one variable. However, when an application has more than one unknown, sometimes it is more convenient to use multiple variables. In this section, we will solve applications containing two unknowns. When two variables are present, the goal is to set up a system of two independent equations. Solving a Cost Application Example 1 At an amusement park, five hot dogs and one drink cost \$16. Two hot dogs and three drinks cost \$9. Find the cost per hot dog and the cost per drink. Solution: Let h represent the cost per hot dog. Label the variables. Let d represent the cost per drink. a cost of 1 Cost of 5 b ϩ a drink b ϭ \$16 hot dogs 5h ϩ d ϭ 16 a cost of 3 Cost of 2 b ϩ a drinks b ϭ \$9 hot dogs 2h ϩ 3d ϭ 9 Write two equations. This system can be solved by either the substitution method or the addition method. We will solve by using the substitution method. The d-variable in the first equation is the easiest variable to isolate. 5h ϩ d ϭ 16 d ϭ Ϫ5h ϩ 16 Solve for d in the first equation. Substitute the quantity Ϫ5h ϩ 16 for d in the second equation. 2h ϩ 3d ϭ 9 2h ϩ 31Ϫ5h ϩ 162 ϭ 9 2h Ϫ 15h ϩ 48 ϭ 9 Clear parentheses. Ϫ13h ϩ 48 ϭ 9 Solve for h. Ϫ13h ϭ Ϫ39 hϭ3 d ϭ Ϫ5132 ϩ 16 dϭ1 Substitute h ϭ 3 in the equation d ϭ Ϫ5h ϩ 16.
27. 27. miL2872X_ch03_177-254 09:22:2006 02:17 PM Section 3.4 IA Page 203 CONFIRMING PAGES 203 Applications of Systems of Linear Equations in Two Variables Because h ϭ 3, the cost per hot dog is \$3.00. Because d ϭ 1, the cost per drink is \$1.00. A word problem can be checked by verifying that the solution meets the conditions specified in the problem. 5 hot dogs ϩ 1 drink ϭ 5(\$3.00) ϩ 1(\$1.00) ϭ \$16.00 as expected 2 hot dogs ϩ 3 drinks ϭ 2(\$3.00) ϩ 3(\$1.00) ϭ \$9.00 as expected Skill Practice 1. At the movie theater, Tom spent \$7.75 on 3 soft drinks and 2 boxes of popcorn. Carly bought 5 soft drinks and 1 box of popcorn for total of \$8.25. Use a system of equations to find the cost of a soft drink and the cost of a box of popcorn. 2. Applications Involving Mixtures Example 2 Solving an Application Involving Chemistry One brand of cleaner used to etch concrete is 25% acid. A stronger industrialstrength cleaner is 50% acid. How many gallons of each cleaner should be mixed to produce 20 gal of a 40% acid solution? Solution: Let x represent the amount of 25% acid cleaner. Let y represent the amount of 50% acid cleaner. 25% Acid 50% Acid 40% Acid Number of gallons of solution x y 20 Number of gallons of pure acid 0.25x 0.50y 0.40(20), or 8 From the first row of the table, we have a total amount amount of Amount of b ϩ a 50% solution b ϭ a of solution b 25% solution x ϩ y ϭ 20 From the second row of the table we have Amount of amount of amount of ° pure acid in ¢ ϩ ° pure acid in ¢ ϭ ° pure acid in ¢ 25% solution 50% solution resulting solution xϩ y ϭ 20 xϩ 0.25x ϩ 0.50y ϭ 8 xϩ y ϭ 20 25x ϩ 50y ϭ 800 y ϭ 20 25x ϩ 50y ϭ 800 Multiply by Ϫ25. 0.25x ϩ 0.50y ϭ 8 Ϫ25x Ϫ 25y ϭ Ϫ500 25x ϩ 50y ϭ 800 25y ϭ 300 y ϭ 12 Multiply by 100 to clear decimals. Create opposite coefficients of x. Add the equations to eliminate x. Skill Practice Answers 1. Soft drink: \$1.25; popcorn: \$2.00
28. 28. miL2872X_ch03_177-254 09:22:2006 204 02:17 PM IA Page 204 CONFIRMING PAGES Chapter 3 Systems of Linear Equations Substitute y ϭ 12 back into one of the original equations. x ϩ y ϭ 20 x ϩ 12 ϭ 20 xϭ8 Therefore, 8 gal of 25% acid solution must be added to 12 gal of 50% acid solution to create 20 gal of a 40% acid solution. Skill Practice 2. A pharmacist needs 8 ounces (oz) of a solution that is 50% saline. How many ounces of 60% saline solution and 20% saline solution must be mixed to obtain the mixture needed? 3. Applications Involving Principal and Interest Example 3 Solving a Mixture Application Involving Finance Serena invested money in two accounts: a savings account that yields 4.5% simple interest and a certificate of deposit that yields 7% simple interest. The amount invested at 7% was twice the amount invested at 4.5%. How much did Serena invest in each account if the total interest at the end of 1 year was \$1017.50? Solution: Let x represent the amount invested in the savings account (the 4.5% account). Let y represent the amount invested in the certificate of deposit (the 7% account). 4.5% Account 7% Account Principal x y Interest 0.045x 0.07y Total 1017.50 Because the amount invested at 7% was twice the amount invested at 4.5%, we have Amount amount ° invested ¢ ϭ 2 ° invested ¢ at 7% at 4.5% y ϭ 2x From the second row of the table, we have Interest interest total ° earned from ¢ ϩ ° earned from ¢ ϭ a interest b 4.5% account 7% account 0.045x ϩ 0.07y ϭ 1017.50 y ϭ 2x 45x ϩ 70y ϭ 1,017,500 Multiply by 1000 to clear decimals. Because the y-variable in the first equation is isolated, we will use the substitution method. Skill Practice Answers 2. 6 oz of 60% solution and 2 oz of 20% solution 45x ϩ 7012x2 ϭ 1,017,500 Substitute the quantity 2x into the second equation.
29. 29. miL2872X_ch03_177-254 09:22:2006 02:17 PM Section 3.4 IA Page 205 CONFIRMING PAGES 205 Applications of Systems of Linear Equations in Two Variables 45x ϩ 140x ϭ 1,017,500 Solve for x. 185x ϭ 1,017,500 xϭ 1,017,500 185 x ϭ 5500 y ϭ 2x y ϭ 2155002 Substitute x ϭ 5500 into the equation y ϭ 2x to solve for y. y ϭ 11,000 Because x ϭ 5500, the amount invested in the savings account is \$5500. Because y ϭ 11,000, the amount invested in the certificate of deposit is \$11,000. Check: \$11,000 is twice \$5500. Furthermore, Interest interest ° earned from ¢ ϩ ° earned from ¢ ϭ \$550010.0452 ϩ \$11,00010.072 ϭ \$1017.50 ✓ 4.5% account 7% account Skill Practice 3. Seth invested money in two accounts, one paying 5% interest and the other paying 6% interest. The amount invested at 5% was \$1000 more than the amount invested at 6%. He earned a total of \$820 interest in 1 year. Use a system of equations to find the amount invested in each account. 4. Applications Involving Distance, Rate, and Time Example 4 Solving a Distance, Rate, and Time Application A plane flies 660 mi from Atlanta to Miami in 1.2 hr when traveling with a tailwind. The return flight against the same wind takes 1.5 hr. Find the speed of the plane in still air and the speed of the wind. Solution: Let p represent the speed of the plane in still air. Let w represent the speed of the wind. The speed of the plane with the wind: (Plane’s still airspeed) ϩ (wind speed): p ϩ w The speed of the plane against the wind: (Plane’s still airspeed) Ϫ (wind speed): p Ϫ w Set up a chart to organize the given information: Distance Rate Time With a tailwind 660 pϩw 1.2 Against a head wind 660 pϪw 1.5 Skill Practice Answers 3. \$8000 invested at 5% and \$7000 invested at 6%
30. 30. miL2872X_ch03_177-254 09:22:2006 206 02:17 PM IA Page 206 CONFIRMING PAGES Chapter 3 Systems of Linear Equations Two equations can be found by using the relationship d ϭ rt 1distance ϭ rate ؒ time2. ° Distance speed time with ¢ ϭ ° with ¢ ° with ¢ wind wind wind 660 ϭ 1p ϩ w2 11.22 Distance speed time ° against ¢ ϭ ° against ¢ ° against ¢ wind wind wind 660 ϭ 1p ϩ w2 11.22 660 ϭ 1p Ϫ w2 11.52 660 ϭ 1p Ϫ w211.52 Notice that the first equation may be divided by 1.2 and still leave integer coefficients. Similarly, the second equation may be simplified by dividing by 1.5. 660 ϭ 1p ϩ w2 11.22 Divide by 1. 2. 660 ϭ 1p Ϫ w2 11.52 Divide by 1.5. 1p ϩ w21.2 660 ϭ 1.2 1.2 550 ϭ p ϩ w 1p Ϫ w21.5 660 ϭ 1.5 1.5 440 ϭ p Ϫ w 550 ϭ p ϩ w 440 ϭ p Ϫ w 990 ϭ 2p Add the equations. p ϭ 495 550 ϭ 14952 ϩ w 55 ϭ w Substitute p ϭ 495 into the equation 550 ϭ p ϩ w. Solve for w. The speed of the plane in still air is 495 mph, and the speed of the wind is 55 mph. Skill Practice 4. A plane flies 1200 mi from Orlando to New York in 2 hr with a tailwind. The return flight against the same wind takes 2.5 hr. Find the speed of the plane in still air and the speed of the wind. 5. Applications Involving Geometry Example 5 Solving a Geometry Application The sum of the two acute angles in a right triangle is 90Њ. The measure of one angle is 6Њ less than 2 times the measure of the other angle. Find the measure of each angle. x Solution: Let x represent the measure of one acute angle. Let y represent the measure of the other acute angle. Skill Practice Answers 4. Speed of plane: 540 mph; speed of wind: 60 mph y
31. 31. miL2872X_ch03_177-254 09:22:2006 02:17 PM Section 3.4 IA Page 207 CONFIRMING PAGES 207 Applications of Systems of Linear Equations in Two Variables The sum of the two acute angles is 90Њ: x ϩ y ϭ 90 One angle is 6Њ less than 2 times the other angle: x ϭ 2y Ϫ 6 x ϩ y ϭ 90 x ϭ 2y Ϫ 6 12y Ϫ 62 ϩ y ϭ 90 3y Ϫ 6 ϭ 90 Because one variable is already isolated, we will use the substitution method. Substitute x ϭ 2y Ϫ 6 into the first equation. 3y ϭ 96 y ϭ 32 x ϭ 2y Ϫ 6 x ϭ 21322 Ϫ 6 To find x, substitute y ϭ 32 into the equation x ϭ 2y Ϫ 6. x ϭ 64 Ϫ 6 x ϭ 58 The two acute angles in the triangle measure 32Њ and 58Њ. Skill Practice 5. Two angles are supplementary. The measure of one angle is 16° less than 3 times the measure of the other. Use a system of equations to find the measures of the angles. Skill Practice Answers 5. 49° and 131° Section 3.4 Practice Exercises Boost your GRADE at mathzone.com! • Practice Problems • Self-Tests • NetTutor • e-Professors • Videos Study Skills Exercise 1. Make up a practice test for yourself. Use examples or exercises from the text. Be sure to cover each concept that was presented. Review Exercises 2. State three methods that can be used to solve a system of linear equations in two variables. For Exercises 3–6, state which method you would prefer to use to solve the system. Then solve the system. 3. y ϭ 9 Ϫ 2x 3x Ϫ y ϭ 16 4. 7x Ϫ y ϭ Ϫ25 2x ϩ 5y ϭ 14 5. 5x ϩ 2y ϭ 6 Ϫ2x Ϫ y ϭ 3 6. x ϭ 5y Ϫ 2 Ϫ3x ϩ 7y ϭ 14 Concept 1: Applications Involving Cost 7. The local community college theater put on a production of Chicago. There were 186 tickets sold, some for \$16 (nonstudent price) and others for \$12 (student price). If the receipts for one performance totaled \$2640, how many of each type of ticket were sold? 8. John and Ariana bought school supplies. John spent \$10.65 on 4 notebooks and 5 pens. Ariana spent \$7.50 on 3 notebooks and 3 pens. What is the cost of 1 notebook and what is the cost of 1 pen?
32. 32. miL2872X_ch03_177-254 208 09:22:2006 02:17 PM Page 208 IA CONFIRMING PAGES Chapter 3 Systems of Linear Equations 9. Joe bought lunch for his fellow office workers on Monday. He spent \$7.35 on 3 hamburgers and 2 fish sandwiches. Corey bought lunch on Tuesday and spent \$7.15 for 4 hamburgers and 1 fish sandwich. What is the price of 1 hamburger, and what is the price of 1 fish sandwich? 10. A group of four golfers pays \$150 to play a round of golf. Of these four, one is a member of the club and three are nonmembers. Another group of golfers consists of two members and one nonmember and pays a total of \$75. What is the cost for a member to play a round of golf, and what is the cost for a nonmember? 11. Meesha has a pocket full of change consisting of dimes and quarters. The total value is \$3.15. There are 7 more quarters than dimes. How many of each coin are there? 12. Crystal has several dimes and quarters in her purse, totaling \$2.70. There is 1 less dime than there are quarters. How many of each coin are there? 13. A coin collection consists of 50¢ pieces and \$1 coins. If there are 21 coins worth \$15.50, how many 50¢ pieces and \$1 coins are there? 14. Suzy has a piggy bank consisting of nickels and dimes. If there are 30 coins worth \$1.90, how many nickels and dimes are in the bank? Concept 2: Applications Involving Mixtures 15. A jar of one face cream contains 18% moisturizer, and another type contains 24% moisturizer. How many ounces of each should be combined to get 12 oz of a cream that is 22% moisturizer? 16. A chemistry student wants to mix an 18% acid solution with a 45% acid solution to get 16 L of a 36% acid solution. How many liters of the 18% solution and how many liters of the 45% solution should be mixed? 17. How much pure bleach must be combined with a solution that is 4% bleach to make 12 oz of a 12% bleach solution? 18. A fruit punch that contains 25% fruit juice is combined with a fruit drink that contains 10% fruit juice. How many ounces of each should be used to make 48 oz of a mixture that is 15% fruit juice? Concept 3: Applications Involving Principal and Interest 19. Alina invested \$27,000 in two accounts: one that pays 2% simple interest and one that pays 3% simple interest. At the end of the first year, her total return was \$685. How much was invested in each account? 20. Didi invested a total of \$12,000 into two accounts paying 7.5% and 6% simple interest. If her total return at the end of the first year was \$840, how much did she invest in each account? 21. A credit union offers 5.5% simple interest on a certificate of deposit (CD) and 3.5% simple interest on a savings account. If Mr. Sorkin invested \$200 more in the CD than in the savings account and the total interest after the first year was \$245, how much was invested in each account? 22. Jody invested \$5000 less in an account paying 4% simple interest than she did in an account paying 3% simple interest. At the end of the first year, the total interest from both accounts was \$675. Find the amount invested in each account. Concept 4: Applications Involving Distance, Rate, and Time 23. It takes a boat 2 hr to go 16 mi downstream with the current and 4 hr to return against the current. Find the speed of the boat in still water and the speed of the current.
33. 33. miL2872X_ch03_177-254 09:22:2006 02:18 PM Section 3.4 Page 209 IA CONFIRMING PAGES 209 Applications of Systems of Linear Equations in Two Variables 24. The Gulf Stream is a warm ocean current that extends from the eastern side of the Gulf of Mexico up through the Florida Straits and along the southeastern coast of the United States to Cape Hatteras, North Carolina. A boat travels with the current 100 mi from Miami, Florida, to Freeport, Bahamas, in 2.5 hr. The return trip against the same current takes 31 hr. Find the speed of the boat in still water and the speed of the current. 3 25. A plane flew 720 mi in 3 hr with the wind. It would take 4 hr to travel the same distance against the wind. What is the rate of the plane in still air and the rate of wind? 26. Nikki and Tatiana rollerblade in opposite directions. Tatiana averages 2 mph faster than Nikki. If they began at the same place and ended up 20 mi apart after 2 hr, how fast did each of them travel? Concept 5: Applications Involving Geometry For Exercises 27–32, solve the applications involving geometry. If necessary, refer to the geometry formulas listed in the inside front cover of the text. 27. In a right triangle, one acute angle measures 6Њ more than 3 times the other. If the sum of the measures of the two acute angles must equal 90Њ, find the measures of the acute angles. 28. An isosceles triangle has two angles of the same measure (see figure). If the angle represented by y measures 3Њ less than the angle x, find the measures of all angles of the triangle. (Recall that the sum of the measures of the angles of a triangle is 180Њ.) yЊ 29. Two angles are supplementary. One angle measures 2Њ less than 3 times the other. What are the measures of the two angles? 30. The measure of one angle is 5 times the measure of another. If the two angles are supplementary, find the measures of the angles. xЊ xЊ 31. One angle measures 3Њ more than twice another. If the two angles are complementary, find the measures of the angles. 32. Two angles are complementary. One angle measures 15° more than 2 times the measure of the other. What are the measures of the two angles? Mixed Exercises 33. How much pure gold (24K) must be mixed with 60% gold to get 20 grams of 75% gold? 34. Two trains leave the depot at the same time, one traveling north and the other traveling south. The speed of one train is 15 mph slower than the other. If after 2 hr the distance between the trains is 190 miles, find the speed of each train. 35. There are two types of tickets sold at the Canadian Formula One Grand Prix race. The price of 6 grandstand tickets and 2 general admissions tickets costs \$2330. The price of 4 grandstand tickets and 4 general admission tickets cost \$2020. What is the price of each type of ticket? 36. A granola mix contains 5% nuts. How many ounces of nuts must be added to get 25 oz of granola with 24% nuts? 37. A bank offers two accounts, a money market account at 2% simple interest and a regular savings account at 1.3% interest. If Svetlana deposits \$3000 between the two accounts and receives \$51.25 total interest in the first year, how much did she invest in each account? 38. A rectangle has the perimeter of 42 m. The length is 1 m longer than the width. Find the dimensions of the rectangle.
34. 34. miL2872X_ch03_177-254 09:22:2006 210 02:18 PM Page 210 IA CONFIRMING PAGES Chapter 3 Systems of Linear Equations 39. Kyle rode his bike for one-half hour. He got a flat tire and had to walk for 1 hr to get home. He rides his bike 2.5 mph faster than he walks. If the distance he traveled was 6.5 miles, what was his speed riding and what was his speed walking? 40. A basketball player scored 19 points by shooting two-point and three-point baskets. If she made a total of eight baskets, how many of each type did she make? 41. In a right triangle, the measure of one acute angle is one-fourth the measure of the other. Find the measures of the acute angles. 42. Angelo invested \$8000 in two accounts: one that pays 3% and one that pays 1.8%. At the end of the first year, his total interest earned was \$222. How much did he deposit in the account that pays 3%? Expanding Your Skills 43. The demand for a certain printer cartridge is related to the price. In general, the higher the price x, the lower the demand y. The supply for the printer cartridges is also related to price. The supply and demand for the printer cartridges depend on the price according to the equations yd ϭ Ϫ10x ϩ 500 20 ys ϭ x 3 where x is the price per cartridge in dollars and yd is the demand measured in 1000s of cartridges Supply/Demand (1000s) For Exercises 43–46, solve the business applications. Supply and Demand of Printer Cartridges Versus Price 500 yd ϭ Ϫ10x ϩ 500 400 ys ϭ 20 x 3 300 200 100 0 0 where x is the price per cartridge in dollars and ys is the supply measured in 1000s of cartridges 10 20 30 40 Price per Cartridge (\$) 50 Find the price at which the supply and demand are in equilibrium (supply ϭ demand), and confirm your answer with the graph. 44. The supply and demand for a pack of note cards depend on the price according to the equations yd ϭ Ϫ130x ϩ 660 where x is the price per pack in dollars and yd is the demand in 1000s of note cards ys ϭ 90x where x is the price per pack in dollars and ys is the supply measured in 1000s of note cards Find the price at which the supply and demand are in equilibrium (supply ϭ demand). 45. A rental car company rents a compact car for \$20 a day, plus \$0.25 per mile. A midsize car rents for \$30 a day, plus \$0.20 per mile. a. Write a linear equation representing the cost to rent the compact car. b. Write a linear equation representing the cost to rent a midsize car. c. Find the number of miles at which the cost to rent either car would be the same. 46. One phone company charges \$0.15 per minute for long-distance calls. A second company charges only \$0.10 per minute for long-distance calls, but adds a monthly fee of \$4.95. a. Write a linear equation representing the cost for the first company. b. Write a linear equation representing the cost for the second company. c. Find the number of minutes of long-distance calling for which the total bill from either company would be the same.
35. 35. miL2872X_ch03_177-254 09:22:2006 Section 3.5 02:18 PM Page 211 IA CONFIRMING PAGES 211 Systems of Linear Equations in Three Variables and Applications Systems of Linear Equations in Three Variables and Applications Section 3.5 1. Solutions to Systems of Linear Equations in Three Variables Concepts In Sections 3.1–3.3, we solved systems of linear equations in two variables. In this section, we will expand the discussion to solving systems involving three variables. A linear equation in three variables can be written in the form Ax ϩ By ϩ Cz ϭ D, where A, B, and C are not all zero. For example, the equation 2x ϩ 3y ϩ z ϭ 6 is a linear equation in three variables. Solutions to this equation are ordered triples of the form (x, y, z) that satisfy the equation. Some solutions to the equation 2x ϩ 3y ϩ z ϭ 6 are Solution: Check: 11, 1, 12 2112 ϩ 3112 ϩ 112 ϭ 6 ✔ True 10, 1, 32 2102 ϩ 3112 ϩ 132 ϭ 6 ✔ True 12, 0, 22 2122 ϩ 3102 ϩ 122 ϭ 6 ✔ True Infinitely many ordered triples serve as solutions to the equation 2x ϩ 3y ϩ z ϭ 6. The set of all ordered triples that are solutions to a linear equation in three variables may be represented graphically by a plane in space. Figure 3-9 shows a portion of the plane 2x ϩ 3y ϩ z ϭ 6 in a 3-dimensional coordinate system. A solution to a system of linear equations in three variables is an ordered triple that satisfies each equation. Geometrically, a solution is a point of intersection of the planes represented by the equations in the system. A system of linear equations in three variables may have one unique solution, infinitely many solutions, or no solution. One unique solution (planes intersect at one point) • The system is consistent. • The system is independent. No solution (the three planes do not all intersect) • The system is inconsistent. • The system is independent. z y x Figure 3-9 1. Solutions to Systems of Linear Equations in Three Variables 2. Solving Systems of Linear Equations in Three Variables 3. Applications of Linear Equations in Three Variables
36. 36. miL2872X_ch03_177-254 212 09:22:2006 02:18 PM IA Page 212 CONFIRMING PAGES Chapter 3 Systems of Linear Equations Infinitely many solutions (planes intersect at infinitely many points) • The system is consistent. • The system is dependent. 2. Solving Systems of Linear Equations in Three Variables To solve a system involving three variables, the goal is to eliminate one variable. This reduces the system to two equations in two variables. One strategy for eliminating a variable is to pair up the original equations two at a time. Solving a System of Three Linear Equations in Three Variables 1. Write each equation in standard form Ax ϩ By ϩ Cz ϭ D. 2. Choose a pair of equations, and eliminate one of the variables by using the addition method. 3. Choose a different pair of equations and eliminate the same variable. 4. Once steps 2 and 3 are complete, you should have two equations in two variables. Solve this system by using the methods from Sections 3.2 and 3.3. 5. Substitute the values of the variables found in step 4 into any of the three original equations that contain the third variable. Solve for the third variable. 6. Check the ordered triple in each of the original equations. Example 1 Solving a System of Linear Equations in Three Variables 2x ϩ y Ϫ 3z ϭ Ϫ7 Solve the system. 3x Ϫ 2y ϩ z ϭ 11 Ϫ2x Ϫ 3y Ϫ 2z ϭ 3 Solution: A 2x ϩ y Ϫ 3z ϭ Ϫ7 B 3x Ϫ 2y ϩ z ϭ 11 C Ϫ2x Ϫ 3y Ϫ 2z ϭ 3 Step 1: The equations are already in standard form. • It is often helpful to label the equations. • The y-variable can be easily eliminated from equations A and B and from equations A and C . This is accomplished by creating opposite coefficients for the y-terms and then adding the equations.
37. 37. miL2872X_ch03_177-254 09:22:2006 02:18 PM Section 3.5 IA Page 213 CONFIRMING PAGES 213 Systems of Linear Equations in Three Variables and Applications Step 2: Eliminate the y-variable from equations A and B . A 2x ϩ y Ϫ 3z ϭ Ϫ7 Multiply by 2. B 3x Ϫ 2y ϩ z ϭ 11 4x ϩ 2y Ϫ 6z ϭ Ϫ14 3x Ϫ 2y ϩ z ϭ 11 Ϫ 5z ϭ Ϫ3 D 7x Step 3: Eliminate the y-variable again, this time from equations A and C . A 2x ϩ y Ϫ 3z ϭ Ϫ7 C Ϫ2x Ϫ 3y Ϫ 2z ϭ Multiply by 3. 6x ϩ 3y Ϫ 9z ϭ Ϫ21 Ϫ2x Ϫ 3y Ϫ 2z ϭ 3 TIP: It is important to note that in steps 2 and 3, the same variable is eliminated. 3 Ϫ 11z ϭ Ϫ18 E 4x Step 4: Now equations D and E can be paired up to form a linear system in two variables. Solve this system. D 7x Ϫ 5z ϭ Ϫ3 E 4x Ϫ 11z ϭ Ϫ18 Multiply by Ϫ4. Multiply by 7. Ϫ28x ϩ 20z ϭ 12 28x Ϫ 77z ϭ Ϫ126 Ϫ57z ϭ Ϫ114 zϭ2 Once one variable has been found, substitute this value into either equation in the two-variable system, that is, either equation D or E . D 7x Ϫ 5z ϭ Ϫ3 7x Ϫ 5122 ϭ Ϫ3 Substitute z ϭ 2 into equation D . 7x Ϫ 10 ϭ Ϫ3 7x ϭ xϭ A 7 1 2x ϩ y Ϫ 3z ϭ Ϫ7 2112 ϩ y Ϫ 3122 ϭ Ϫ7 2 ϩ y Ϫ 6 ϭ Ϫ7 y Ϫ 4 ϭ Ϫ7 y ϭ Ϫ3 The solution is (1, Ϫ3, 2). Check: Step 5: Now that two variables are known, substitute these values for x and z into any of the original three equations to find the remaining variable y. Substitute x ϭ 1 and z ϭ 2 into equation A . Step 6: Check the ordered triple in the three original equations. 2x ϩ y Ϫ 3z ϭ Ϫ7 3x Ϫ 2y ϩ z ϭ 11 Ϫ2x Ϫ 3y Ϫ 2z ϭ 3 2112 ϩ 1Ϫ32 Ϫ 3122 ϭ Ϫ7 ✔ True 3112 Ϫ 21Ϫ32 ϩ 122 ϭ 11 ✔ True Ϫ2112 Ϫ 31Ϫ32 Ϫ 2122 ϭ 3 ✔ True Skill Practice 1. Solve the system. x ϩ 2y ϩ z ϭ 1 3x Ϫ y ϩ 2z ϭ 13 2x ϩ 3y Ϫ z ϭ Ϫ8 Skill Practice Answers 1. (1, Ϫ2, 4)
38. 38. miL2872X_ch03_177-254 214 09:22:2006 02:18 PM IA Page 214 CONFIRMING PAGES Chapter 3 Systems of Linear Equations Example 2 Applying Systems of Linear Equations in Three Variables In a triangle, the smallest angle measures 10Њ more than one-half of the largest angle. The middle angle measures 12Њ more than the smallest angle. Find the measure of each angle. Solution: zЊ Let x represent the measure of the smallest angle. yЊ xЊ Let y represent the measure of the middle angle. Let z represent the measure of the largest angle. To solve for three variables, we need to establish three independent relationships among x, y, and z. z ϩ 10 2 A xϭ B y ϭ x ϩ 12 The smallest angle measures 10° more than onehalf the measure of the largest angle. The middle angle measures 12Њ more than the measure of the smallest angle. C x ϩ y ϩ z ϭ 180 The sum of the interior angles of a triangle measures 180°. Clear fractions and write each equation in standard form. Standard Form A x y B C z ϭ ϩ 10 2 Multiply by 2. 2x ϭ z ϩ 20 Ϫx ϩ y ϭ x ϩ 12 xϩyϩzϭ 2x Ϫ z ϭ 20 ϭ 12 x ϩ y ϩ z ϭ 180 180 Notice equation B is missing the z-variable. Therefore, we can eliminate z again by pairing up equations A and C . A 2x C Ϫ z ϭ 120 x ϩ y ϩ z ϭ 180 3x ϩ y ϭ 200 B Ϫx ϩ y ϭ 12 D 3x ϩ y ϭ 200 D Multiply by Ϫ1. x Ϫ y ϭ Ϫ12 3x ϩ y ϭ 200 4x ϭ 188 x ϭ 47 Pair up equations B and D to form a system of two variables. Solve for x.
39. 39. miL2872X_ch03_177-254 09:22:2006 02:18 PM Section 3.5 IA Page 215 CONFIRMING PAGES 215 Systems of Linear Equations in Three Variables and Applications From equation B we have Ϫx ϩ y ϭ 12 From equation C we have Ϫ47 ϩ y ϭ 12 47 ϩ 59 ϩ z ϭ 180 x ϩ y ϩ z ϭ 180 y ϭ 59 z ϭ 74 The smallest angle measures 47Њ, the middle angle measures 59Њ, and the largest angle measures 74Њ. Skill Practice 2. The perimeter of a triangle is 30 in. The shortest side is 4 in. shorter than the longest side. The longest side is 6 in. less than the sum of the other two sides. Find the length of each side. Example 3 Solving a Dependent System of Linear Equations Solve the system. If there is not a unique solution, label the system as either dependent or inconsistent. A 3x ϩ y Ϫ z ϭ 8 B 2x Ϫ y ϩ 2z ϭ 3 x ϩ 2y Ϫ 3z ϭ 5 C Solution: The first step is to make a decision regarding the variable to eliminate. The y-variable is particularly easy to eliminate because the coefficients of y in equations A and B are already opposites. The y-variable can be eliminated from equations B and C by multiplying equation B by 2. A 3x ϩ y Ϫ z ϭ 8 Pair up equations A and B to eliminate y. B 2x Ϫ y ϩ 2z ϭ 3 5x ϩ z ϭ 11 D B 2x Ϫ y ϩ 2z ϭ 3 C Multiply by 2. x ϩ 2y Ϫ 3z ϭ 5 4x Ϫ 2y ϩ 4z ϭ 6 x ϩ 2y Ϫ 3z ϭ 5 ϩ z ϭ 11 E 5x Pair up equations B and C to eliminate y. Because equations D and E are equivalent equations, it appears that this is a dependent system. By eliminating variables we obtain the identity 0 ϭ 0. D 5x ϩ z ϭ 11 E 5x ϩ z ϭ 11 Multiply by Ϫ1. Ϫ5x Ϫ z ϭ Ϫ11 5x ϩ z ϭ 11 0ϭ 0 The result 0 ϭ 0 indicates that there are infinitely many solutions and that the system is dependent. Skill Practice 3. Solve the system. If the system does not have a unique solution, identify the system as dependent or inconsistent. xϩ yϩ zϭ 8 Skill Practice Answers 2x Ϫ y ϩ z ϭ 6 2. 8 in., 10 in., and 12 in. 3. Dependent system Ϫ5x Ϫ 2y Ϫ 4z ϭ Ϫ30
40. 40. miL2872X_ch03_177-254 09:22:2006 216 02:18 PM IA Page 216 CONFIRMING PAGES Chapter 3 Systems of Linear Equations Example 4 Solving an Inconsistent System of Linear Equations Solve the system. If there is not a unique solution, identify the system as either dependent or inconsistent. 2x ϩ 3y Ϫ 7z ϭ 4 Ϫ4x Ϫ 6y ϩ 14 z ϭ 1 5x ϩ y Ϫ 3z ϭ 6 Solution: We will eliminate the x-variable. Multiply by 2. A 2x ϩ 3y Ϫ 7z ϭ 4 B Ϫ4x Ϫ 6y ϩ 14 z ϭ 1 Ϫ4x Ϫ 6y ϩ 14 z ϭ 1 C 5x ϩ y Ϫ 3z ϭ 6 0ϭ9 4x ϩ 6y Ϫ 14z ϭ 8 (contradiction) The result 0 ϭ 9 is a contradiction, indicating that the system has no solution. The system is inconsistent. Skill Practice 4. Solve the system. If the system does not have a unique solution, identify the system as dependent or inconsistent. x Ϫ 2y ϩ z ϭ 5 x Ϫ 3y ϩ 2z ϭ Ϫ7 Ϫ2x ϩ 4y Ϫ 2z ϭ 6 3. Applications of Linear Equations in Three Variables Example 5 Applying Systems of Linear Equations to Nutrition Doctors have become increasingly concerned about the sodium intake in the U.S. diet. Recommendations by the American Medical Association indicate that most individuals should not exceed 2400 mg of sodium per day. Liz ate 1 slice of pizza, 1 serving of ice cream, and 1 glass of soda for a total of 1030 mg of sodium. David ate 3 slices of pizza, no ice cream, and 2 glasses of soda for a total of 2420 mg of sodium. Melinda ate 2 slices of pizza, 1 serving of ice cream, and 2 glasses of soda for a total of 1910 mg of sodium. How much sodium is in one serving of each item? Solution: Let x represent the sodium content of 1 slice of pizza. Let y represent the sodium content of 1 serving of ice cream. Let z represent the sodium content of 1 glass of soda. x ϩ y ϩ z ϭ 1030 From Liz’s meal we have: A Skill Practice Answers From David’s meal we have: B 3x 4. Inconsistent system From Melinda’s meal we have: C 2x ϩ ϩ 2z ϭ 2420
41. 41. miL2872X_ch03_177-254 09:22:2006 02:18 PM Section 3.5 IA Page 217 CONFIRMING PAGES 217 Systems of Linear Equations in Three Variables and Applications Equation B is missing the y-variable. Eliminating y from equations A and C , we have A x ϩ y ϩ z ϭ 1030 Multiply by Ϫ1. Ϫx Ϫ y Ϫ z ϭ Ϫ1030 C 2x ϩ y ϩ 2z ϭ 1910 2x ϩ y ϩ 2z ϭ 1910 ϩ zϭ 880 D x Solve the system formed by equations B and D . 3x ϩ 2z ϭ B 3x ϩ 2z ϭ 2420 D x ϩ z ϭ 880 Multiply by Ϫ2. 2420 Ϫ2x Ϫ 2z ϭ Ϫ1760 x From equation D we have x ϩ z ϭ 880 From equation A we have x ϩ y ϩ z ϭ 1030 ϭ 660 660 ϩ z ϭ 880 z ϭ 220 660 ϩ y ϩ 220 ϭ 1030 y ϭ 150 Therefore, 1 slice of pizza has 660 mg of sodium, 1 serving of ice cream has 150 mg of sodium, and 1 glass of soda has 220 mg of sodium. Skill Practice 5. Annette, Barb, and Carlita work in a clothing shop. One day the three had combined sales of \$1480. Annette sold \$120 more than Barb. Barb and Carlita combined sold \$280 more than Annette. How much did each person sell? Section 3.5 Skill Practice Answers 5. Annette sold \$600, Barb sold \$480, and Carlita sold \$400. Practice Exercises Boost your GRADE at mathzone.com! • Practice Problems • Self-Tests • NetTutor • e-Professors • Videos Study Skills Exercises 1. Look back over your notes for this chapter. Have you highlighted the important topics? Have you underlined the key terms? Have you indicated the places where you are having trouble? If you find that you have problems with a particular topic, write a question that you can ask your instructor either in class or in the office. 2. Define the key terms. a. Linear equation in three variables b. Ordered triple Review Exercises For Exercises 3–4, solve the systems by using two methods: (a) the substitution method and (b) the addition method. 3. 3x ϩ y ϭ 4 4x ϩ y ϭ 5 4. 2x Ϫ 5y ϭ 3 Ϫ4x ϩ 10y ϭ 3
42. 42. miL2872X_ch03_177-254 09:22:2006 218 02:18 PM IA Page 218 CONFIRMING PAGES Chapter 3 Systems of Linear Equations 5. Two cars leave Kansas City at the same time. One travels east and one travels west. After 3 hr the cars are 369 mi apart. If one car travels 7 mph slower than the other, find the speed of each car. Concept 1: Solutions to Systems of Linear Equations in Three Variables 6. How many solutions are possible when solving a system of three equations with three variables? 7. Which of the following points are solutions to the system? 8. Which of the following points are solutions to the system? 12, 1, 72, 13, Ϫ10, Ϫ62, 14, 0, 22 11, 1, 32, 10, 0, 42, 14, 2, 12 2x Ϫ y ϩ z ϭ 10 4x ϩ 2y Ϫ 3z ϭ 10 x Ϫ 3y ϩ 2z ϭ 8 Ϫ3x Ϫ 3y Ϫ 6z ϭ Ϫ24 Ϫ9x Ϫ 6y ϩ 3z ϭ Ϫ45 9x ϩ 3y Ϫ 9z ϭ 33 9. Which of the following points are solutions to the system? 10. Which of the following points are solutions to the system? 112, 2, Ϫ22, 14, 2, 12, 11, 1, 12 10, 4, 32, 13, 6, 102, 13, 3, 12 x ϩ 2y Ϫ z ϭ 5 x Ϫ 3y ϩ z ϭ Ϫ5 Ϫ2x ϩ y Ϫ z ϭ Ϫ4 Ϫx Ϫ y Ϫ 4z ϭ Ϫ6 x Ϫ 3y ϩ z ϭ Ϫ1 4x ϩ y Ϫ z ϭ 4 Concept 2: Solving Systems of Linear Equations in Three Variables For Exercises 11–24, solve the system of equations. 11. 2x ϩ y Ϫ 3z ϭ Ϫ12 3x Ϫ 2y Ϫ z ϭ 3 Ϫx ϩ 5y ϩ 2z ϭ Ϫ3 14. 6x Ϫ 5y ϩ z ϭ 7 5x ϩ 3y ϩ 2z ϭ 0 Ϫ2x ϩ y Ϫ 3z ϭ 11 17. 4x ϩ 2z ϭ 12 ϩ 3y 2y ϭ 3x ϩ 3z Ϫ 5 y ϭ 2x ϩ 7z ϩ 8 20. x Ϫ y Ϫ z ϭ Ϫ11 x ϩ y Ϫ z ϭ 15 2x Ϫ y ϩ z ϭ Ϫ9 ϭ 8 23. 4x ϩ 9y 8x ϩ 6z ϭ Ϫ1 6y ϩ 6z ϭ Ϫ1 12. Ϫ3x Ϫ 2y ϩ 4z ϭ Ϫ15 2x ϩ 5y Ϫ 3z ϭ 3 4x Ϫ y ϩ 7z ϭ 15 13. x Ϫ 3y Ϫ 4z ϭ Ϫ7 5x ϩ 2y ϩ 2z ϭ Ϫ1 4x Ϫ y Ϫ 5z ϭ Ϫ6 15. Ϫ3x ϩ y Ϫ z ϭ 8 Ϫ4x ϩ 2y ϩ 3z ϭ Ϫ3 2x ϩ 3y Ϫ 2z ϭ Ϫ1 16. 2x ϩ 3y ϩ 3z ϭ 15 3x Ϫ 6y Ϫ 6z ϭ Ϫ23 Ϫ9x Ϫ 3y ϩ 6z ϭ 8 19. xϩ yϩzϭ 6 Ϫx ϩ y Ϫ z ϭ Ϫ2 2x ϩ 3y ϩ z ϭ 11 22. x ϩ y ϩ 3z ϭ 2 2x Ϫ 3z ϭ 5 3y ϩ 3z ϭ 2 18. y ϭ 2x ϩ z ϩ 1 Ϫ3x Ϫ 1 ϭ Ϫ2y ϩ 2z 5x ϩ 3z ϭ 16 Ϫ 3y 21. 2x Ϫ 3y ϩ 2z ϭ Ϫ1 x ϩ 2y ϭ Ϫ4 x ϩ zϭ 1 ϩ 2z ϭ 11 y Ϫ 7z ϭ 4 x Ϫ 6y ϭ 1 24. 3x Concept 3: Applications of Linear Equations in Three Variables 25. A triangle has one angle that measures 5Њ more than twice the smallest angle, and the largest angle measures 11Њ less than 3 times the measure of the smallest angle. Find the measures of the three angles.
43. 43. miL2872X_ch03_177-254 09:22:2006 Section 3.5 02:18 PM Page 219 IA CONFIRMING PAGES 219 Systems of Linear Equations in Three Variables and Applications 26. The largest angle of a triangle measures 4Њ less than 5 times the measure of the smallest angle. The middle angle measures twice that of the smallest angle. Find the measures of the three angles. 27. The perimeter of a triangle is 55 cm. The measure of the shortest side is 8 cm less than the middle side. The measure of the longest side is 1 cm less than the sum of the other two sides. Find the lengths of the sides. 28. The perimeter of a triangle is 5 ft. The longest side of the triangle measures 20 in. more than the shortest side. The middle side is 3 times the measure of the shortest side. Find the lengths of the three sides in inches. 29. A movie theater charges \$7 for adults, \$5 for children under age 17, and \$4 for seniors over age 60. For one showing of Batman the theater sold 222 tickets and took in \$1383. If twice as many adult tickets were sold as the total of children and senior tickets, how many tickets of each kind were sold? 30. Goofie Golf has 18 holes that are par 3, par 4, or par 5. Most of the holes are par 4. In fact, there are 3 times as many par 4s as par 3s. There are 3 more par 5s than par 3s. How many of each type are there? 31. Combining peanuts, pecans, and cashews makes a party mixture of nuts. If the amount of peanuts equals the amount of pecans and cashews combined, and if there are twice as many cashews as pecans, how many ounces of each nut is used to make 48 oz of party mixture? 32. Souvenir hats, T-shirts, and jackets are sold at a rock concert. Three hats, two T-shirts, and one jacket cost \$140. Two hats, two T-shirts, and two jackets cost \$170. One hat, three T-shirts, and two jackets cost \$180. Find the prices of the individual items. 33. In 2002, Baylor University in Waco, Texas, had twice as many students as Vanderbilt University in Nashville, Tennessee. Pace University in New York City had 2800 more students than Vanderbilt University. If the enrollment for all three schools totaled 27,200, find the enrollment for each school. 34. Annie and Maria traveled overseas for seven days and stayed in three different hotels in three different cities: Stockholm, Sweden; Oslo, Norway; and Paris, France. The total bill for all seven nights (not including tax) was \$1040. The total tax was \$106. The nightly cost (excluding tax) to stay at the hotel in Paris was \$80 more than the nightly cost (excluding tax) to stay in Oslo. Find the cost per night for each hotel excluding tax. Number of Nights Cost/Night (\$) Tax Rate Paris, France 1 x 8% Stockholm, Sweden 4 y 11% Oslo, Norway 2 z 10% City Mixed Exercises For Exercises 35–44, solve the system. If there is not a unique solution, label the system as either dependent or inconsistent. 35. 2x ϩ y ϩ 3z ϭ 2 x Ϫ y ϩ 2z ϭ Ϫ4 x ϩ 3y Ϫ z ϭ 1 38. 3x ϩ 2y ϩ z ϭ 3 x Ϫ 3y ϩ z ϭ 4 Ϫ6x Ϫ 4y Ϫ 2z ϭ 1 36. x ϩyϭ z 3(x Ϫ y) ϩ 6z ϭ 1 Ϫ y 7x ϩ 3(y ϩ 1) ϭ 7 Ϫ z ϭ 5 39. 1x ϩ 2y 2 3 2 1 3 1 x Ϫ 2z ϭ Ϫ10 5 3 1 1 4 3y Ϫ 4z ϭ 37. 6x Ϫ 2y ϩ 2z ϭ 2 4x ϩ 8y Ϫ 2z ϭ 5 Ϫ2x Ϫ 4y ϩ z ϭ Ϫ2 40. 1x ϩ 1y ϩ z ϭ 3 2 4 1 x ϩ 1y ϩ 1z ϭ 9 8 4 4 8 x Ϫ y Ϫ 2z ϭ 1 3 3
44. 44. miL2872X_ch03_177-254 09:22:2006 02:18 PM 220 IA Page 220 CONFIRMING PAGES Chapter 3 Systems of Linear Equations 2x ϩ y ϭ Ϫ3 2y ϩ 16z ϭ Ϫ10 Ϫ7x Ϫ 3y ϩ 4z ϭ 8 41. 2x ϩ y ϭ 31z Ϫ 12 3x Ϫ 21y Ϫ 2z2 ϭ 1 212x Ϫ 3z2 ϭ Ϫ6 Ϫ 2y 42. 43. Ϫ0.1y ϩ 0.2z ϭ 0.2 0.1x ϩ 0.1y ϩ 0.1z ϭ 0.2 Ϫ0.1x ϩ 0.3z ϭ 0.2 ϭ 0 44. 0.1x Ϫ 0.2y 0.3y ϩ 0.1z ϭ Ϫ0.1 0.4x Ϫ 0.1z ϭ 1.2 Expanding Your Skills The systems in Exercises 45–48 are called homogeneous systems because each system has (0, 0, 0) as a solution. However, if a system is dependent, it will have infinitely many more solutions. For each system determine whether (0, 0, 0) is the only solution or if the system is dependent. 45. 2x Ϫ 4y ϩ 8z ϭ 0 Ϫx Ϫ 3y ϩ z ϭ 0 x Ϫ 2y ϩ 5z ϭ 0 Section 3.6 Concepts 1. Introduction to Matrices 2. Solving Systems of Linear Equations by Using the Gauss-Jordan Method 46. 2x Ϫ 4y ϩ z ϭ 0 x Ϫ 3y Ϫ z ϭ 0 3x Ϫ y ϩ 2z ϭ 0 47. 4x Ϫ 2y Ϫ 3z ϭ 0 Ϫ8x Ϫ y ϩ z ϭ 0 3 2x Ϫ y Ϫ z ϭ 0 2 48. 5x ϩ y ϭ0 4y Ϫ z ϭ 0 5x ϩ 5y Ϫ z ϭ 0 Solving Systems of Linear Equations by Using Matrices 1. Introduction to Matrices In Sections 3.2, 3.3, and 3.5, we solved systems of linear equations by using the substitution method and the addition method. We now present a third method called the Gauss-Jordan method that uses matrices to solve a linear system. A matrix is a rectangular array of numbers (the plural of matrix is matrices). The rows of a matrix are read horizontally, and the columns of a matrix are read vertically. Every number or entry within a matrix is called an element of the matrix. The order of a matrix is determined by the number of rows and number of columns. A matrix with m rows and n columns is an m ϫ n (read as “m by n”) matrix. Notice that with the order of a matrix, the number of rows is given first, followed by the number of columns. Example 1 1 d 27 Determining the Order of a Matrix Determine the order of each matrix. 2 a. c 5 Ϫ4 p 1.9 0 ¥ b. ≥ 7.2 Ϫ6.1 1 c. £ 0 0 0 1 0 0 0§ 1 d. 3a b c4 Solution: a. This matrix has two rows and three columns. Therefore, it is a 2 ϫ 3 matrix. b. This matrix has four rows and one column. Therefore, it is a 4 ϫ 1 matrix. A matrix with one column is called a column matrix.
45. 45. miL2872X_ch03_177-254 09:22:2006 02:18 PM Section 3.6 IA Page 221 CONFIRMING PAGES 221 Solving Systems of Linear Equations by Using Matrices c. This matrix has three rows and three columns. Therefore, it is a 3 ϫ 3 matrix. A matrix with the same number of rows and columns is called a square matrix. d. This matrix has one row and three columns. Therefore, it is a 1 ϫ 3 matrix. A matrix with one row is called a row matrix. Skill Practice 1. Ϫ5 £ 1 8 Determine the order of the matrix. 2. 34 Ϫ 84 2 3§ 9 5 £ 10 § 15 3. 4. c 2 Ϫ1 Ϫ0.5 d 6 A matrix can be used to represent a system of linear equations written in standard form. To do so, we extract the coefficients of the variable terms and the constants within the equation. For example, consider the system 2x Ϫ y ϭ 5 x ϩ 2y ϭ Ϫ5 The matrix A is called the coefficient matrix. Aϭ c 2 1 Ϫ1 d 2 If we extract both the coefficients and the constants from the equations, we can construct the augmented matrix of the system: c 2 1 5 Ϫ1 d ` 2 Ϫ5 A vertical bar is inserted into an augmented matrix to designate the position of the equal signs. Example 2 Writing the Augmented Matrix of a System of Linear Equations Write the augmented matrix for each linear system. a. Ϫ3x Ϫ 4y ϭ 3 Ϫ 3z ϭ 14 b. 2x 2x ϩ 4y ϭ 2 2y ϩ z ϭ 2 xϩ y ϭ 4 Solution: a. c Ϫ3 Ϫ4 3 ` d 2 4 2 2 b. £ 0 1 0 2 1 TIP: Notice that zeros are inserted to denote the coefficient of each missing term. Ϫ3 14 1 † 2§ 0 4 Skill Practice 5. Ϫx ϩ y ϭ 4 2x Ϫ y ϭ 1 Skill Practice Answers Write the augmented matrix for the system. 6. 2x Ϫ y ϩ z ϭ 14 Ϫ3x ϩ 4y ϭ 8 x Ϫ y ϩ 5z ϭ 0 1. 3 ϫ 2 2. 3. 3 ϫ 1 4. Ϫ1 1 ` 5. c 2 Ϫ1 2 Ϫ1 6. £Ϫ3 4 1 Ϫ1 1ϫ2 2ϫ2 4 d 1 1 14 0 † 8§ 5 0
46. 46. miL2872X_ch03_177-254 09:22:2006 222 02:18 PM IA Page 222 CONFIRMING PAGES Chapter 3 Systems of Linear Equations Writing a Linear System from an Augmented Matrix Example 3 Write a system of linear equations represented by each augmented matrix. a. c 2 4 1 c. £ 0 0 2 b. £ 1 3 Ϫ5 Ϫ8 d ` 6 1 0 1 0 Ϫ1 1 1 3 14 Ϫ2 † Ϫ5 § Ϫ1 2 0 4 0 † Ϫ1 § 1 0 Solution: a. 2x Ϫ 5y ϭ Ϫ8 4x ϩ y ϭ b. 2x Ϫ y ϩ 3z ϭ 14 x ϩ y Ϫ 2z ϭ Ϫ5 6 3x ϩ y Ϫ z ϭ x ϩ 0y ϩ 0z ϭ c. 0x ϩ y ϩ 0z ϭ Ϫ1 0x ϩ 0y ϩ z ϭ Skill Practice 7. c 2 Ϫ1 xϭ 4 or 4 y ϭ Ϫ1 zϭ 0 2 0 Write a system of linear equations represented by each augmented matrix. 3 5 ` d 8 1 Ϫ3 8. £ 14 Ϫ8 2 1 3 1 4 0 † 20 § 5 6 1 9. £ 0 0 0 1 0 0 Ϫ5 0 † 2§ 1 0 2. Solving Systems of Linear Equations by Using the Gauss-Jordan Method We know that interchanging two equations results in an equivalent system of linear equations. Interchanging two rows in an augmented matrix results in an equivalent augmented matrix. Similarly, because each row in an augmented matrix represents a linear equation, we can perform the following elementary row operations that result in an equivalent augmented matrix. Elementary Row Operations The following elementary row operations performed on an augmented matrix produce an equivalent augmented matrix: 1. Interchange two rows. 2. Multiply every element in a row by a nonzero real number. 3. Add a multiple of one row to another row. Skill Practice Answers 7. 2x ϩ 3y ϭ 5 Ϫx ϩ 8y ϭ 1 8. Ϫ3x ϩ 2y ϩ z ϭ 4 14x ϩ y ϭ 20 Ϫ8x ϩ 3y ϩ 5z ϭ 6 9. x ϭ Ϫ5, y ϭ 2, z ϭ 0 When we are solving a system of linear equations by any method, the goal is to write a series of simpler but equivalent systems of equations until the solution is obvious. The Gauss-Jordan method uses a series of elementary row operations performed on the augmented matrix to produce a simpler augmented matrix. In
47. 47. miL2872X_ch03_177-254 09:22:2006 02:18 PM Section 3.6 IA Page 223 CONFIRMING PAGES 223 Solving Systems of Linear Equations by Using Matrices particular, we want to produce an augmented matrix that has 1s along the diagonal of the matrix of coefficients and 0s for the remaining entries in the matrix of coefficients. A matrix written in this way is said to be written in reduced row echelon form. For example, the augmented matrix from Example 3(c) is written in reduced row echelon form. 1 £0 0 0 1 0 0 4 0 † Ϫ1 § 1 0 The solution to the corresponding system of equations is easily recognized as x ϭ 4, y ϭ Ϫ1, and z ϭ 0. Similarly, matrix B represents a solution of x ϭ a and y ϭ b. Bϭ c Example 4 0 a ` d 1 b 1 0 Solving a System of Linear Equations by Using the Gauss-Jordan Method Solve by using the Gauss-Jordan method. 2x Ϫ y ϭ 5 x ϩ 2y ϭ Ϫ5 Solution: c 2 1 5 Ϫ1 d ` 2 Ϫ5 Set up the augmented matrix. c 1 2 2 Ϫ5 d ` 5 Ϫ1 Switch row 1 and row 2 to get a 1 in the upper left position. Ϫ2R1 ϩ R2 1 R2 c 1 0 2 Ϫ5 d ` Ϫ5 15 Multiply row 1 by Ϫ2 and add the result to row 2. This produces an entry of 0 below the upper left position. Ϫ1R2 1 R2 5 c 1 0 2 Ϫ5 ` d 1 Ϫ3 Multiply row 2 by Ϫ1 to 5 produce a 1 along the diagonal in the second row. R1 3 R2 Ϫ2R2 ϩ R1 1 R1 c 1 0 Multiply row 2 by Ϫ2 and add the result to row 1. This produces a 0 in the first row, second column. 1 0 d ` 1 Ϫ3 The matrix C is in reduced row echelon form. From the augmented matrix, we have x ϭ 1 and y ϭ Ϫ3. The solution to the system is (1, Ϫ3). Cϭ c 1 0 0 1 d ` 1 Ϫ3 Skill Practice 10. Solve by using the Gauss-Jordan method. x Ϫ 2y ϭ Ϫ21 2x ϩ y ϭ Ϫ2 Skill Practice Answers 10. (Ϫ5, 8)
48. 48. miL2872X_ch03_177-254 224 09:22:2006 02:18 PM IA Page 224 CONFIRMING PAGES Chapter 3 Systems of Linear Equations The order in which we manipulate the elements of an augmented matrix to produce reduced row echelon form was demonstrated in Example 4. In general, the order is as follows. • First produce a 1 in the first row, first column. Then use the first row to obtain 0s in the first column below this element. • Next, if possible, produce a 1 in the second row, second column. Use the second row to obtain 0s above and below this element. • Next, if possible, produce a 1 in the third row, third column. Use the third row to obtain 0s above and below this element. • The process continues until reduced row echelon form is obtained. Example 5 Solving a System of Linear Equations by Using the Gauss-Jordan Method Solve by using the Gauss-Jordan method. ϭ Ϫy ϩ 5 x Ϫ2x ϩ 2z ϭ y Ϫ 10 3x ϩ 6y ϩ 7z ϭ 14 Solution: First write each equation in the system in standard form. x Ϫ2x ϩ ϭ Ϫy ϩ 5 2z ϭ y Ϫ 10 xϩ y 3x ϩ 6y ϩ 7z ϭ 1 £ Ϫ2 3 Ϫ1R2 ϩ R1 1 R1 Ϫ3R2 ϩ R3 1 R3 2R3 ϩ R1 1 R1 Ϫ2R3 ϩ R2 1 R2 5 Ϫ2x Ϫ y ϩ 2z ϭ Ϫ10 3x ϩ 6y ϩ 7z ϭ 14 2R1 ϩ R2 1 R2 Ϫ3R1 ϩ R3 1 R3 ϭ 1 Ϫ1 6 14 0 5 2 † Ϫ10 § 7 14 Set up the augmented matrix. 1 £0 0 1 1 3 0 5 2 † 0§ 7 Ϫ1 Multiply row 1 by 2 and add the result to row 2. Multiply row 1 by Ϫ3 and add the result to row 3. 1 £0 0 0 1 0 Ϫ2 5 2 † 0§ 1 Ϫ1 Multiply row 2 by Ϫ1 and add the result to row 1. Multiply row 2 by Ϫ3 and add the result to row 3. 1 £0 0 0 1 0 0 3 0 † 2§ 1 Ϫ1 Multiply row 3 by 2 and add the result to row 1. Multiply row 3 by Ϫ2 and add the result to row 2. From the reduced row echelon form of the matrix, we have x ϭ 3, y ϭ 2, and z ϭ Ϫ1. The solution to the system is (3, 2, Ϫ1).
49. 49. miL2872X_ch03_177-254 09:22:2006 02:18 PM Section 3.6 Skill Practice IA Page 225 CONFIRMING PAGES 225 Solving Systems of Linear Equations by Using Matrices Solve by using the Gauss-Jordan method. 11. x ϩ y ϩ z ϭ 2 xϪ yϩ zϭ4 x ϩ 4y ϩ 2z ϭ 1 It is particularly easy to recognize a dependent or inconsistent system of equations from the reduced row echelon form of an augmented matrix. This is demonstrated in Examples 6 and 7. Example 6 Solving a Dependent System of Equations by Using the Gauss-Jordan Method Solve by using the Gauss-Jordan method. x Ϫ 3y ϭ 4 1 3 xϪ yϭ2 2 2 Solution: 1 c1 c Ϫ1R1 ϩ R2 1 R2 2 Set up the augmented matrix. 2 Ϫ3 4 ` d Ϫ3 2 2 1 0 Ϫ3 4 ` d 0 0 Multiply row 1 by Ϫ1 and add the 2 result to row 2. The second row of the augmented matrix represents the equation 0 ϭ 0; hence, the system is dependent. The solution is 51x, y2 0 x Ϫ 3y ϭ 46. Skill Practice Solve by using the Gauss-Jordan method. 12. 4x Ϫ 6y ϭ 16 6x Ϫ 9y ϭ 24 Example 7 Solving an Inconsistent System of Equations by Using the Gauss-Jordan Method Solve by using the Gauss-Jordan method. x ϩ 3y ϭ 2 Ϫ3x Ϫ 9y ϭ 1 Solution: c 3R1 ϩ R2 1 R2 1 3 2 ` d Ϫ3 Ϫ9 1 c 1 0 3 2 ` d 0 7 Set up the augmented matrix. Multiply row 1 by 3 and add the result to row 2. The second row of the augmented matrix represents the contradiction 0 ϭ 7; hence, the system is inconsistent. There is no solution. Skill Practice Answers 11. (1,Ϫ1, 2) 12. Infinitely many solutions; 51x, y2 0 4x Ϫ 6y ϭ 166; dependent system
50. 50. miL2872X_ch03_177-254 09:22:2006 226 02:18 PM IA Page 226 CONFIRMING PAGES Chapter 3 Systems of Linear Equations Skill Practice 13. Solve by using the Gauss-Jordan method. 6x ϩ 10y ϭ 1 15x ϩ 25y ϭ 3 Skill Practice Answers 13. No solution; Inconsistent system Calculator Connections Many graphing calculators have a matrix editor in which the user defines the order of the matrix and then enters the elements of the matrix. For example, the 2 ϫ 3 matrix Dϭ c 2 3 Ϫ3 Ϫ13 d ` 8 1 is entered as shown. Once an augmented matrix has been entered into a graphing calculator, a rref function can be used to transform the matrix into reduced row echelon form. Section 3.6 Practice Exercises Boost your GRADE at mathzone.com! • Practice Problems • Self-Tests • NetTutor • e-Professors • Videos Study Skills Exercises 1. Prepare a one-page summary sheet with the most important information that you need for the next test. On the day of the test, look at this sheet several times to refresh your memory, instead of trying to memorize new information. 2. Define the key terms. a. Matrix b. Order of a matrix c. Column matrix d. Square matrix e. Row matrix f. Coefficient matrix g. Augmented matrix h. Reduced row echelon form
51. 51. miL2872X_ch03_177-254 09:22:2006 02:18 PM Section 3.6 IA Page 227 CONFIRMING PAGES 227 Solving Systems of Linear Equations by Using Matrices Review Exercises For Exercises 3–5, solve the system by using any method. 3. x Ϫ 6y ϭ 9 4. x ϩ y Ϫ z ϭ 8 5. 2x Ϫ y ϩ z ϭ Ϫ4 x ϩ 2y ϭ 13 x Ϫ 2y ϩ z ϭ 3 Ϫx ϩ y ϩ 3z ϭ Ϫ7 x ϩ 3y ϩ 2z ϭ 7 x ϩ 3y Ϫ 4z ϭ 22 Concept 1: Introduction to Matrices For Exercises 6–14, (a) determine the order of each matrix and (b) determine if the matrix is a row matrix, a column matrix, a square matrix, or none of these. 4 5 ¥ 6. ≥ Ϫ3 0 3 Ϫ1 Ϫ9 d Ϫ3 5 12. c 4.3 Ϫ8.1 Ϫ9 9. c Ϫ9 8. £ Ϫ1 5 5 7. £ Ϫ1 § 2 11. 30 10. 34 Ϫ74 13. c 4.2 0 d 18 3 1 3 3 4 Ϫ2 1 Ϫ8 5 14. £ Ϫ1 0 6 d Ϫ7 8 4 Ϫ8 8 11 3 4§ 7 54 1 2§ 7 For Exercises 15–19, set up the augmented matrix. x Ϫ 2y ϭ Ϫ1 15. 16. 2x ϩ y ϭ Ϫ7 x Ϫ 2y 18. x Ϫ 3y ϭ 3 17. y ϭ 2x Ϫ 1 y ϭ Ϫ3x ϩ 7 2x Ϫ 5y ϭ 4 ϭ5Ϫz 19. 5x Ϫ 17 ϭ Ϫ2z 2x ϩ 6y ϩ 3z ϭ Ϫ2 8x 3x Ϫ y Ϫ 2z ϭ 8x ϩ 3y Ϫ 12z ϭ 1 ϩ 6z ϭ 26 ϩ y 24 For Exercises 20–23, write a system of linear equations represented by the augmented matrix. 20. c 4 12 6 3 d ` 5 Ϫ6 21. c Ϫ2 Ϫ7 Ϫ15 5 d ` Ϫ45 15 1 22. £ 0 0 0 1 0 0 4 0 † Ϫ1 § 1 7 1 23. £ 0 0 0 1 0 0 0.5 0 † 6.1 § 1 3.9 Concept 2: Solving Systems of Linear Equations by Using the Gauss-Jordan Method 25. Given the matrix F 24. Given the matrix E Eϭ c 3 9 8 Ϫ2 ` d 7 Ϫ1 Fϭ c 1 12 0 8 d ` Ϫ2 Ϫ13 a. What is the element in the second row and third column? a. What is the element in the second row and second column? b. What is the element in the first row and second column? b. What is the element in the first row and third column?
52. 52. miL2872X_ch03_177-254 09:22:2006 02:19 PM 228 IA Page 228 CONFIRMING PAGES Chapter 3 Systems of Linear Equations 27. Given the matrix J 26. Given the matrix Z Zϭ c 2 2 Jϭ c 11 1 d ` 1 Ϫ1 29. Given the matrix L 28. Given the matrix K 5 1 Lϭ c 1 2 ` d 3 Ϫ4 30. Given the matrix M 1 Ϫ3 13 9 6 d ` 19 Ϫ7 2 write the matrix obtained by interchanging rows 1 and 2. write the matrix obtained by interchanging rows 1 and 2. Mϭ c 7 1 d ` 3 Ϫ6 write the matrix obtained by multiplying the elements in the second row by 1. 3 write the matrix obtained by multiplying the elements in the first row by 1. 2 Kϭ c 1 0 31. Given the matrix N 2 5 d ` Ϫ4 Ϫ1 Nϭ c write the matrix obtained by multiplying the first row by 3 and adding the result to row 2. 1 3 Ϫ5 d ` 12 Ϫ2 2 write the matrix obtained by multiplying the first row by 2 and adding the result to row 2. 32. Given the matrix R 33. Given the matrix S 1 3 0 Ϫ1 R ϭ £ 4 1 Ϫ5 † 6§ Ϫ2 0 Ϫ3 10 1 2 0 10 S ϭ £ 5 1 Ϫ4 † 3 § Ϫ3 4 5 2 a. Write the matrix obtained by multiplying the first row by Ϫ4 and adding the result to row 2. a. Write the matrix obtained by multiplying the first row by Ϫ5 and adding the result to row 2. b. Using the matrix obtained from part (a), write the matrix obtained by multiplying the first row by 2 and adding the result to row 3. b. Using the matrix obtained from part (a), write the matrix obtained by multiplying the first row by 3 and adding the result to row 3. For Exercises 34–49, solve the systems by using the Gauss-Jordan method. 34. x Ϫ 2y ϭ Ϫ1 35. 2x ϩ y ϭ Ϫ7 38. x ϩ 3y ϭ 3 39. x ϩ 3y ϭ Ϫ1 Ϫ3x Ϫ 6y ϭ 12 2x ϩ 5y ϭ 1 xϩ yϭ 4 2x Ϫ 4y ϭ Ϫ4 37. 2x Ϫ 3y ϭ Ϫ2 x ϩ 2y ϭ 13 44. xϪyϭ4 41. 2x Ϫ y ϭ 0 2x ϩ y ϭ 5 40. Ϫ4x Ϫ 10y ϭ Ϫ2 43. x ϩ 3y ϭ 6 Ϫ4x Ϫ 9y ϭ 3 36. 2x Ϫ 5y ϭ 4 4x ϩ 12y ϭ 12 42. x Ϫ 3y ϭ 3 xϩyϭ3 3x ϩ y ϭ Ϫ4 Ϫ6x Ϫ 2y ϭ 3 45. 2x ϩ y ϭ 4 6x ϩ 3y ϭ Ϫ1
53. 53. miL2872X_ch03_177-254 09:22:2006 02:19 PM Section 3.6 46. x ϩ y ϩ z ϭ 6 IA Page 229 CONFIRMING PAGES 229 Solving Systems of Linear Equations by Using Matrices 47. 2x Ϫ 3y Ϫ 2z ϭ 11 xϪyϩzϭ2 x ϩ 3y ϩ 8z ϭ xϩyϪzϭ0 1 3x Ϫ y ϩ 14z ϭ Ϫ2 48. x Ϫ 2y ϭ5Ϫz 49. 5x Ϫ 10z ϭ 15 2x ϩ 6y ϩ 3z ϭ Ϫ10 x Ϫ y ϩ 6z ϭ 23 3x Ϫ y Ϫ 2z ϭ x ϩ 3y Ϫ 12z ϭ 13 5 For Exercises 50– 53, use the augmented matrices A, B, C, and D to answer true or false. Aϭ c 6 5 2 Ϫ4 ` d 7 Ϫ2 Bϭ c 5 6 7 Ϫ2 ` d 2 Ϫ4 Cϭ c 1 1 Ϫ2 3 ` 3d 5 Ϫ2 7 Dϭ c 7 5 Ϫ2 d ` Ϫ12 8 Ϫ4 50. The matrix A is a 2 ϫ 3 matrix. 51. Matrix B is equivalent to matrix A. 52. Matrix A is equivalent to matrix C. 53. Matrix B is equivalent to matrix D. 54. What does the notation R2 3 R1 mean when one is performing the Gauss-Jordan method? 55. What does the notation 2R3 1 R3 mean when one is performing the Gauss-Jordan method? 56. What does the notation Ϫ3R1 ϩ R2 1 R2 mean when one is performing the Gauss-Jordan method? 57. What does the notation 4R2 ϩ R3 1 R3 mean when one is performing the Gauss-Jordan method? Graphing Calculator Exercises For Exercises 58–63, use the matrix features on a graphing calculator to express each augmented matrix in reduced row echelon form. Compare your results to the solution you obtained in the indicated exercise. 58. c 1 2 Ϫ1 Ϫ2 d ` Ϫ7 1 Compare with Exercise 34. 61. c 2 1 Ϫ2 Ϫ3 d ` 13 2 Compare with Exercise 37. 59. c 1 2 3 Ϫ3 ` d 4 Ϫ5 Compare with Exercise 35. 1 62. £ 1 1 1 1 6 Ϫ1 1 † 2§ 1 Ϫ1 0 Compare with Exercise 46. 60. c 6 1 3 ` d 3 Ϫ4 Ϫ9 Compare with Exercise 36. 2 Ϫ3 Ϫ2 11 3 8 † 1§ 63. £ 1 3 Ϫ1 14 Ϫ2 Compare with Exercise 47.
54. 54. miL2872X_ch03_177-254 09:22:2006 230 02:19 PM IA Page 230 CONFIRMING PAGES Chapter 3 Systems of Linear Equations Section 3.7 Concepts 1. Introduction to Determinants 2. Determinant of a 3 ؋ 3 Matrix 3. Cramer’s Rule Determinants and Cramer’s Rule 1. Introduction to Determinants Associated with every square matrix is a real number called the determinant of the matrix. A determinant of a square matrix A, denoted det A, is written by enclosing the elements of the matrix within two vertical bars. For example, If A ϭ c If 0 Bϭ £ 4 Ϫ2 2 6 Ϫ1 d 0 Ϫ5 0 10 1 1 2§ 1 then det A ϭ ` 2 6 0 then det B ϭ † 4 Ϫ2 Ϫ1 ` 0 Ϫ5 0 10 1 1 2 1 † Determinants have many applications in mathematics, including solving systems of linear equations, finding the area of a triangle, determining whether three points are collinear, and finding an equation of a line between two points. The determinant of a 2 ϫ 2 matrix is defined as follows: Determinant of a 2 ϫ 2 Matrix a c b d is the real number ad Ϫ bc. It is written as d a c b ` ϭ ad Ϫ bc d The determinant of the matrix c ` Example 1 Evaluating a 2 ؋ 2 Determinant Evaluate the determinants. a. ` 6 5 Ϫ2 1 3 ` b. ` 2 0 Ϫ11 ` 0 Solution: a. ` 6 5 Ϫ2 1 3 ` For this determinant, a ϭ 6, b ϭ Ϫ2, c ϭ 5, and d ϭ 1 . 3 1 ad Ϫ bc ϭ 162a b Ϫ 1Ϫ22152 3 ϭ 2 ϩ 10 ϭ 12 TIP: Example 1(b) illustrates that the value of a determinant having a row of all zeros is 0. The same is true for a determinant having a column of all zeros. b. ` 2 0 Ϫ11 ` 0 For this determinant, a ϭ 2, b ϭ Ϫ11, c ϭ 0, d ϭ 0. ad Ϫ bc ϭ 122 102 Ϫ 1Ϫ112102 ϭ0Ϫ0 ϭ0
55. 55. miL2872X_ch03_177-254 09:22:2006 02:19 PM IA Page 231 Section 3.7 231 Determinants and Cramer’s Rule Evaluate the determinants. Skill Practice 1a. ` CONFIRMING PAGES 2 8 ` Ϫ1 5 b. ` Ϫ6 0 ` 4 0 2. Determinant of a 3 ؋ 3 Matrix To find the determinant of a 3 ϫ 3 matrix, we first need to define the minor of an element of the matrix. For any element of a 3 ϫ 3 matrix, the minor of that element is the determinant of the 2 ϫ 2 matrix obtained by deleting the row and column in which the element resides. For example, consider the matrix 5 £0 4 Ϫ1 Ϫ7 2 6 1§ 6 The minor of the element 5 is found by deleting the first row and first column and then evaluating the determinant of the remaining 2 ϫ 2 matrix: 5 £0 4 Ϫ1 Ϫ7 2 6 1§ 6 Now evaluate the determinant: ` Ϫ7 2 1 ` ϭ (Ϫ7)(6) Ϫ (1)(2) 6 ϭ Ϫ44 For this matrix, the minor for the element 5 is Ϫ44. To find the minor of the element Ϫ7, delete the second row and second column, and then evaluate the determinant of the remaining 2 ϫ 2 matrix. 5 £0 4 Ϫ1 Ϫ7 2 6 1§ 6 Now evaluate the determinant: ` 5 4 6 ` ϭ (5)(6) Ϫ (6)(4) ϭ 6 6 For this matrix, the minor for the element Ϫ7 is 6. Example 2 Determining the Minor for Elements in a 3 ؋ 3 Matrix Find the minor for each element in the first column of the matrix. 3 £2 0 Ϫ1 5§ Ϫ6 4 Ϫ4 1 Solution: For 3: 3 £2 0 4 Ϫ4 1 Ϫ1 5§ Ϫ6 The minor is: ` Ϫ4 1 5 ` ϭ (Ϫ4)(Ϫ6) Ϫ (5)(1) ϭ 19 Ϫ6 Skill Practice Answers 1a. 18 b. 0
56. 56. miL2872X_ch03_177-254 09:22:2006 232 02:19 PM IA Page 232 CONFIRMING PAGES Chapter 3 Systems of Linear Equations For 2: 3 £2 0 4 Ϫ4 1 Ϫ1 5§ Ϫ6 The minor is: For 0: 3 £2 0 4 Ϫ4 1 Ϫ1 5§ Ϫ6 The minor is: ` Ϫ1 ` ϭ (4)(Ϫ6) Ϫ (Ϫ1)(1) ϭ Ϫ23 Ϫ6 4 1 ` 4 Ϫ4 Ϫ1 ` ϭ (4)(5) Ϫ (Ϫ1)(Ϫ4) ϭ 16 5 Skill Practice 2. Find the minor for the element 3. £ Ϫ1 1 2 5 8 3 7 Ϫ6 2§ 4 The determinant of a 3 ϫ 3 matrix is defined as follows. Definition of a Determinant of a 3 ؋ 3 Matrix a1 † a2 a3 b1 b2 b3 c1 b c2 † ϭ a1 ؒ ` 2 b3 c3 c2 b ` Ϫ a2 ؒ ` 1 c3 b3 c1 b ` ϩ a3 ؒ ` 1 c3 b2 c1 ` c2 From this definition, we see that the determinant of a 3 ϫ 3 matrix can be written as a1ؒ(minor of a1) Ϫ a2ؒ(minor of a2) ϩ a3ؒ(minor of a3) Evaluating determinants in this way is called expanding minors. Example 3 Evaluating a 3 ؋ 3 Determinant 2 Evaluate the determinant. † 1 Ϫ5 4 Ϫ3 5 2 0† Ϫ1 Solution: 2 † 1 Ϫ5 4 Ϫ3 5 2 Ϫ3 0† ϭ2ؒ ` 5 Ϫ1 4 0 ` Ϫ 112 ؒ ` 5 Ϫ1 2 4 ` ϩ 1Ϫ52 ؒ ` Ϫ1 Ϫ3 ϭ 2 3 1Ϫ32 1Ϫ12 Ϫ 102 152 4 Ϫ 1 3 142 1Ϫ12 Ϫ 122 152 4 Ϫ 5 3 142 102 Ϫ 1221Ϫ32 4 ϭ 2132 Ϫ 11Ϫ142 Ϫ 5162 ϭ 6 ϩ 14 Ϫ 30 ϭ Ϫ10 Skill Practice Answers 2. ` Ϫ1 Ϫ6 ` ϭ 26 5 4 2 ` 0