1. A system of linear equations is a group of equations that are considered at the same time. The solution of a system of linear equations is the set of ordered pairs that make all equations in the system true.
2. Solve the system of equations by graphing. x – y = 5 x + 2y = -4 x – y = 5 5 – 0 =5 (5, 0) 0 –(-5) = 5 (0, -5) x + 2y = -4 0 + 2(-2) = -4 (0, -2) -4 + 2(0) = -4 (-4, 0) The solution is consistent and independent The solution appears to be (2, -3) Example 1-1a
3. Check the solution x – y = 5 x + 2y = -4 x – y = 5 2 – -3 =5 2 + 3 = 5 5 = 5 x + 2y = -4 2 + 2(-3) = -4 2 – 6 = -4 -4 = -4 Example 1-1a
4. Solve the system of equations by graphing. y = 2x – 3 y = -2x + 5 The solution appears to be (2, 1) y = 2x – 3 1 = 2(2) – 3 1 = 4 – 3 1 = 1 y = -2x + 5 1 = -2(2) + 5 1 = -4 + 5 1 = 1 The solution is consistent and independent Example 1-1a
5. Solve the system of equations by graphing. 0 – 2(0) = 0 (0, 0) 2 – 2(1) = 0 (2, 1) 3 + 3 = 6 (3, 3) 1 + 5 = 6 (1, 5) The solution is consistent and independent The solution appears to be (4, 2) Example 1-1a
6. Original equations Replace x with 4and y with 2. Simplify. Check Substitute the coordinates into each equation. Answer: The solution of the system is (4, 2). Example 1-1a
7. What is the solution for the system of equations? Parallel lines No solution The system is inconsistent. Example 1-5a
8. Graph the system of equations and describe it as consistent and independent, consistent and dependent, or inconsistent. Write the equations in slope-intercept form. Infinite solutions The system is consistent and dependent. Example 1-4a
10. Let Fund-raisingA service club is selling copies of their holiday cookbook to raise funds for a project. The printer’s set-up charge is $200, and each book costs $2 to print. The cookbooks will sell for $6 each. How many cookbooks must the members sell before they make a profit? What is the independent variable? Cookbooks What is the dependent variable? Dollars Example 1-2a
11. price per book number of books. is times $ of income Let $ to produce is cost per book plus set-up charge. Fund-raisingA service club is selling copies of their holiday cookbook to raise funds for a project. The printer’s set-up charge is $200, and each book costs $2 to print. The cookbooks will sell for $6 each. How many cookbooks must the members sell before they make a profit? y = 2x + 200 y = 6 x Example 1-2a
12. Solve the system: y = 2x + 200 y = 6x The graphsintersect at (50, 300). This is the break-evenpoint. If the groupsells less than 50 books, they will lose money.If the groupsellsmore than 50 books, they will make aprofit. Example 1-2a
13. Use substitution to solve the system of equations. x = 26 – 4y x = 26 – 4y x = 26 – 4(4) x – 5y = -10 x = 26 – 16 26 – 4y – 5y = -10 x = 10 26 – 9y = -10 – 9y = -36 y = 4 ( , 4) 10 Example 2-1a
14. Use substitution to solve the system of equations. x = 3y + 2 x = 3(1) + 2 x = 3 + 2 3y + 2 + 7y = 12 x = 5 10y + 2 = 12 10y = 10 y = 1 ( , 1) 5 Example 2-1b
15. Use elimination to solve the system 3 2x + 3y = 1 2x + 3(1) = 1 2x + 3 = 1 2x = -2 x = -1 6x + 9y = 3 -6x - 8y = -2 2x + 3y = 1 3x + 4y = 1 -2 y = 1 ( , 1 ) -1 Example 2-2b
16. Use the elimination method to solve the system of equations. Multiply by 2. Use multiplication to eliminate x. False statement No solution Lines are parallel Example 2-5a
17. Use elimination to solve the system -2 -8x - 2y = -6 8x + 2y = 6 4x + y = 3 8x + 2y = 6 0 = 0 true statement lines coincide All real numbers are solutions Example 2-2b
18. Abigale bought 46 dozen roses for her floral shop. Red roses cost $15 per dozen and white roses cost $18 per dozen. If she spent a total of $765 for all of the roses, how many dozen of each color rose did she purchase? R = dozens of red roses purchased W = dozens of white roses purchased R + W = 46 15R + 18W = 765 -15R + -15W = -690 15R + 18W = 765 3W = 75 W = 25 R = 21
