1. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Section 2.4 Solving Linear Equations: ax  b  c

2. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Objective o Solve equations of the form ax + b = c.

3. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Solving Equations of the Form ax + b = c Procedure for Solving Linear Equations that Simplify to the Form ax  b  c 1.Combine like terms on both sides of the equation. 2.Use the addition principle of equality and add the opposite of the constant b to both sides. 3.Use the multiplication (or division) principle of equality to multiply both sides by the reciprocal of the coefficient of the variable (or divide both sides by the coefficient itself). The coefficient of the variable will become +1.

4. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Solving Equations of the Form ax + b = c Procedure for Solving Linear Equations that Simplify to the Form ax  b  c (cont.) 4.Check your answer by substituting it for the variable in the original equation.

5. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 1: Solving Linear Equations Solve each of the following equations. a.3x + 3 =  18 Solution Write the equation. Add  3 to both sides. Simplify. Divide both sides by 3. Simplify.

6. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 1: Solving Linear Equations (cont.) Check: Substitute x =  7. Simplify. True statement

7. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 1: Solving Linear Equations (cont.) b. –26 = 2y – 14 – 4y Solution Write the equation. Combine like terms. Add 14 to both sides. Simplify. Divide both sides by  2. Simplify.

8. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 1: Solving Linear Equations (cont.) Check: Substitute y = 6. Simplify. True statement

9. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 2: Solving Linear Equations Involving Decimals Solve each of the following equations. a.16.53 – 18.2z – 7.43 = 0 Solution Write the equation. Multiply both sides by 100. (This results in integer coefficients.) Simplify. Combine like terms. Add  910 to both sides. Simplify.

10. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 2: Solving Linear Equations Involving Decimals (cont.) Check: Divide both sides by  1820. Simplify. Substitute z = 0.5. Simplify. True statement

11. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 2: Solving Linear Equations Involving Decimals (cont.) b. 5.1x + 7.4 – 1.8x =  9.1 Solution Write the equation. Multiply both sides by 10. (This results in integer coefficients.) Simplify. Combine like terms. Add  74 to both sides.

12. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 2: Solving Linear Equations Involving Decimals (cont.) Divide both sides by the coefficient 33. Simplify.

13. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 3: Solving Linear Equations with Fractional Coefficients Solve each of the following equations. Solution Apply the distributive property. Multiply both sides by 18 (the LCM of the denominators). Write the equation.

14. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 3: Solving Linear Equations with Fractional Coefficients (cont.) Simplify. Divide both sides by 15. Add 45 to both sides. Simplify.

15. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 3: Solving Linear Equations with Fractional Coefficients (cont.) Check: Substitute Simplify. True statement

16. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 3: Solving Linear Equations with Fractional Coefficients (cont.) Solution Write the equation. Multiply both sides by 12 (the LCM of the denominators). Apply the distributive property.

17. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 3: Solving Linear Equations with Fractional Coefficients (cont.) Simplify. Combine like terms. Add  42 to both sides. Simplify. Divide both sides by 7. Simplify. Checking will show that  6 is the solution.

18. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Solving Equations of the Form ax + b = c Notes ABOUT CHECKING Checking can be quite time-consuming and need not be done for every problem. This is particularly important on exams. You should check only if you have time after the entire exam is completed.

19. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Practice Problems Solve the following linear equations. 1. x + 14 – 8x = −7 2. 2.4 = 2.6y – 5.9y – 0.9 3. 4.

20. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Practice Problem Answers 1.  x = 3 2. y =  13. n = 2 4. x =  7