1. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Section 2.5 Solving Linear Equations: ax  b  cx  d

2. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Objectives o Solve equations of the form ax  b  cx  d. o Understand the terms conditional equations, identities, and contradictions.

3. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Solve Equations of the Form ax + b = cx + d General Procedure for Solving Linear Equations that Simplify to the Form ax + b = cx + d 1.Simplify each side of the equation by removing any grouping symbols and combining like terms on both sides of the equation. 2.Use the addition principle of equality and add the opposite of a constant term and/or variable term to both sides so that variables are on one side and constants are on the other side.

4. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Solve Equations of the Form ax + b = cx + d General Procedure for Solving Linear Equations that Simplify to the Form ax + b = cx + d (cont.) 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.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 Equations with Variables on Both Sides Solve the following equations. Solution

6. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 1: Solving Equations with Variables on Both Sides (cont.) Check:

7. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 1: Solving Equations with Variables on Both Sides (cont.) Solution

8. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 1: Solving Equations with Variables on Both Sides (cont.) Check:

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 Solution

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:

11. 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 Solution

12. 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.)

13. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 4: Solving Equations with Parentheses Solve the following equations. Solution

14. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 4: Solving Equations with Parentheses (cont.)

15. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 4: Solving Equations with Parentheses (cont.) Solution

16. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 4: Solving Equations with Parentheses (cont.)

17. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 5: Solutions of Equations Determine whether each of the following equations is a conditional equation, an identity, or a contradiction. Solution The equation has one solution and it is a conditional equation.

18. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 5: Solutions of Equations (cont.) Solution The last equation is never true. Therefore, the original equation is a contradiction and has no solution.

19. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 5: Solutions of Equations (cont.) Solution The last equation is always true. Therefore, the original equation is an identity and has an infinite number of solutions. Every real number is a solution.

20. 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. Determine whether each of the following equations is a conditional equation, an identity, or a contradiction.

21. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Practice Problem Answers 1. x  32. x   1 3. x   2 4. n   75. 6. Identity 7. Contradiction