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Chapter 3

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Chapter 3

1. 1. Chapter 3 Solving Linear Equations
2. 2. 3.1 Solving Equations Using Addition and Subtraction <ul><li>You can solve an equation by using the transformations below to isolate the variable on one side of the equation. </li></ul><ul><li>When you rewrite an equation using these transformations, you produce an equation with the same solutions as the original equation. </li></ul><ul><li>These equations are called equivalent equations. </li></ul>
3. 3. <ul><li>To change, or transform , an equation into an equivalent equation, think of an equation as having two sides that are “in balance.” </li></ul><ul><li>Any transformation you apply to an equation must keep the equation in balance. </li></ul><ul><li>For example, if you if you subtract 3 from one side of the equation, you must also subtract 3 from the other side of the equation. </li></ul>
4. 4. Transformations that produce Equivalent Equations 7 = x Interchange the sides x = 8 – 3 Simplify one or both sides x + 6 = 10 Subtract the same # to each side x – 3 = 5 Add the same # to each side Equivalent Equation Original Equation
5. 5. <ul><li>Example 1 : Solve. </li></ul><ul><li>a) x – 5 = -13 </li></ul><ul><li>b) x – 9 = -17 </li></ul><ul><li>Example 2 : Solve. </li></ul><ul><li>a) -8 = n – (-4) </li></ul><ul><li>b) -11 = n – (-2) </li></ul>
6. 6. <ul><li>Linear Equations: The variable is raised to the first power and does not occur in a denominator, inside a square root, or inside absolute value symbols. </li></ul>| x + 3| = 7 - 4 + n = 2 n – 6 x 2 + 5 = 9 x + 5 = 9 Not a Linear Equation Linear Equation
7. 7. <ul><li>Example 3 : Several record temperature changes have taken place in Spearfish, South Dakota. On January 22, 1943, the temperature in Spearfish fell from 54 °F at 9:00am to -4°F at 9:27am. By how many degrees did the temperature fall? </li></ul>
8. 8. <ul><li>Example 4 : Match the real-life problem with an equation. </li></ul><ul><li>x – 4 = 16 x + 16 = 4 16 – x = 4 </li></ul><ul><li>a) You owe \$16 to your cousin. You paid x dollars back and now you owe \$4. How much did you pay back? </li></ul><ul><li>b) The temperature was x °F. It rose 16°F and is now 4°F. What was the original temperature? </li></ul><ul><li>c) A telephone pole extends 4 feet below ground and 16 feet above ground. What is the total length x of the pole? </li></ul>
9. 9. 3.2 Solving Equations using Multiplication and Division 4 x = 12 Divide each equation by the same nonzero # Multiply each equation by the same nonzero # Equivalent Equation Original Equation
10. 10. <ul><li>Example 1 : Solve. </li></ul><ul><li>a) - 4 x = 1 </li></ul><ul><li>b) 7 n = - 35 </li></ul><ul><li>Example 2 : Solve. </li></ul><ul><li>a) </li></ul><ul><li>b) </li></ul>
11. 11. <ul><li>Example 3 : Solve. </li></ul><ul><li>a) </li></ul><ul><li>b) </li></ul>
12. 12. <ul><li>Properties of Equality : </li></ul><ul><li>Addition Property: </li></ul><ul><li>Subtraction Property: </li></ul><ul><li>Multiplication Property: </li></ul><ul><li>Division Property: </li></ul>
13. 13. 3.3 Solving Multi-Step Equations <ul><li>Solving a linear equation may require two or more transformations. </li></ul><ul><li>Simplify one or both sides of the equation (if needed). </li></ul><ul><li>Use the inverse operations to isolate the variable. </li></ul>
14. 14. <ul><li>Example 1 : Solve. </li></ul><ul><li>a) </li></ul><ul><li>b) </li></ul><ul><li>Example 2 : Solve. </li></ul><ul><li>a) 7 x – 3 x – 8 = 24 </li></ul><ul><li>b) 2 x – 9 x + 17 = - 4 </li></ul>
15. 15. <ul><li>Example 3 : Solve. </li></ul><ul><li>a) 5 x + 3( x + 4) = 28 </li></ul><ul><li>b) 4 x – 3( x – 2) = 21 </li></ul><ul><li>Example 4 : Solve. </li></ul><ul><li>a) 4 x + 12( x – 3) = 28 </li></ul><ul><li>b) 2 x – 5( x – 9) = 27 </li></ul>
16. 16. <ul><li>Example 5 : Solve. </li></ul><ul><li>a) </li></ul><ul><li>b) </li></ul><ul><li>Example 6 : A body temperature of 95 °F or lower may indicate the medical condition called hypothermia. What temperature in the Celsius scale may indicate hypothermia? </li></ul>
17. 17. <ul><li>Example 7 : The temperature within Earth’s crust increases about 30 ° Celsius for each kilometer of depth beneath the surface. If the temperature at Earth’s surface is 24°C, at what depth would you expect the temperature to be 114°C? </li></ul>
18. 18. 3.6 <ul><li>Objective: To find exact and apporoximate solutions of equations that contain decimals. </li></ul><ul><li>Round-off Error : </li></ul><ul><li>Example 1 : Solve the equation. Round to the nearest hundredth. </li></ul><ul><li>a) 7.23x + 16.51 = 47.89 – 2.55x </li></ul>
19. 19. <ul><li>Example 1 : Solve the equation. Round to the nearest hundredth. </li></ul><ul><li>a) 7.23x + 16.51 = 47.89 – 2.55x </li></ul><ul><li>b) 6.6(1.2 – 7.3x) = 16.4x + 5.8 </li></ul>
20. 20. <ul><li>Example 2 : Multiply the equation by a power of 10 to write an equivalent equation with integer coefficients. Solve the equation and round to the nearest hundredth. </li></ul><ul><li>a) 3.11x – 17.64 = 2.02x -5.89 </li></ul><ul><li>b) 5.8 + 3.2x = 3.4x – 16.7 </li></ul>
21. 21. 3.7 Formulas and Functions <ul><li>Objective: To solve a formula for one of its variables and rewrite an equation in function form </li></ul><ul><li>Formula : an algebraic expression that relates two or more real-life quantities. </li></ul>
22. 22. <ul><li>Example 1 : Use the formula for area of a rectangle – A = lw </li></ul><ul><li>a) Find a formula for l in terms of A and w </li></ul><ul><li>b) Use the new formula to find the length of a rectangle that has an area of 35 sq. ft. and a width of 7 feet. </li></ul>
23. 23. <ul><li>Example 2 : Solve the temperature formula C = 5/9(F – 32) for F . </li></ul><ul><li>Example 3 : a) Solve the simple interest formula for r . </li></ul><ul><li>b) Find the interest rate for an investment of \$1500 that earned \$54 in simple interest in one year </li></ul>
24. 24. <ul><li>Example 3 : Rewrite the equation </li></ul><ul><li>3x + y = 4 so that y is a function of x. </li></ul><ul><li>Example 4 : a) Rewrite the equation </li></ul><ul><li>3x + y = 4 so that x is a function of y. </li></ul><ul><li>b) Use the result to find x when y = -2, -1, 0 and 1 </li></ul>