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Copyright © 2010 Pearson Education, Inc. All rights reserved. 2.1 – Slide 1 Chapter 2 Equations, Inequalities, and Applications Copyright © 2010 Pearson Education, Inc. All rights reserved. 2.1 – Slide 2 2.1 The Addition Property of Equality Copyright © 2010 Pearson Education, Inc. All rights reserved. 2.1 – Slide 3 2.1 The Addition Property of Equality Objectives 1. 2. 3. Identify linear equations. Use the addition property of equality. Simplify, and then use the addition property of equality. Copyright © 2010 Pearson Education, Inc. All rights reserved. 2.1 – Slide 4 2.1 The Addition Property of Equality Identifying Linear Equations Linear Equation in One Variable A linear equation in one variable can be written in the form Ax + B = C where A, B, and C are real numbers, with A ≠ 0. Some examples of linear and nonlinear equations follow. 4x + 9 = 0, x2 + 2x = 5, 2x – 3 = 5, and x=7 Linear 1 = 6, and |2x + 6| = 0 Nonlinear x Copyright © 2010 Pearson Education, Inc. All rights reserved. 2.1 – Slide 5 2.1 The Addition Property of Equality Identifying Linear Equations A solution of an equation is a number that makes the equation true when it replaces the variable. Equations that have exactly the same solution sets are equivalent equations. A linear equation is solved by using a series of steps to produce a simpler equivalent equation of the form x = a number or a number = x. Copyright © 2010 Pearson Education, Inc. All rights reserved. 2.1 – Slide 6 2.1 The Addition Property of Equality Using the Addition Property of Equality Addition Property of Equality If A, B, and C are real numbers, then the equations A=B and A+C=B+C are equivalent equations. In words, we can add the same number to each side of an equation without changing the solution. Copyright © 2010 Pearson Education, Inc. All rights reserved. 2.1 – Slide 7 2.1 The Addition Property of Equality Using the Addition Property of Equality Note Equations can be thought of in terms of a balance. Thus, adding the same quantity to each side does not affect the balance. Copyright © 2010 Pearson Education, Inc. All rights reserved. 2.1 – Slide 8 2.1 The Addition Property of Equality Using the Addition Property of Equality Example 1 Solve each equation. Our goal is to get an equivalent equation of the form x = a number. (a) x – 23 = 8 x – 23 + 23 = 8 + 23 x = 31 Check: 31 – 23 = 8 Copyright © 2010 Pearson Education, Inc. All rights reserved. (b) y – 2.7 = –4.1 y – 2.7 + 2.7 = –4.1 + 2.7 y = – 1.4 Check: –1.4 – 2.7 = –4.1 2.1 – Slide 9 2.1 The Addition Property of Equality Using the Addition Property of Equality The same number may be subtracted from each side of an equation without changing the solution. If a is a number and –x = a, then x = –a. Copyright © 2010 Pearson Education, Inc. All rights reserved. 2.1 – Slide 10 2.1 The Addition Property of Equality Using the Addition Property of Equality Example 2 Solve each equation. Our goal is to get an equivalent equation of the form x = a number. (a) –12 = z + 5 –12 – 5 = z + 5 – 5 –17 = z Check: –12 = –17 + 5 (b) 4a + 8 = 3a 4a – 4a + 8 = 3a – 4a 8 = –a –8 = a Check: 4(–8) + 8 = 3(–8) ? –24 = –24 Copyright © 2010 Pearson Education, Inc. All rights reserved. 2.1 – Slide 11 2.1 The Addition Property of Equality Simplifying and Using the Addition Property of Equality Check: 5((2 · –36) –3) – (11(–36) + 1) = Example 3 Solve. 5(2b – 3) – (11b + 1) = 20 10b – 15 – 11b – 1 = 20 –b – 16 = 20 5(–72 –3) – (–396 + 1) = 5(–75) – (–395) = –375 + 395 = 20 –b – 16 + 16 = 20 + 16 –b = 36 b = –36 Copyright © 2010 Pearson Education, Inc. All rights reserved. 2.1 – Slide 12