Chapter 2 Equations and Inequalities in One Variable

Chapter 2 Equations and Inequalities in One Variable Section 1 Linear Equations: The Addition and Multiplication Properties of Equality

Section 2. 1 Objectives 1 Determine Whether a Number Is a Solution of an Equation 2 Use the Addition Property of Equality to Solve Linear Equations 3 Use the Multiplication Property of Equality to Solve Linear Equations Copyright © 2014, 2010, 2007 Pearson Education, Inc. 2

Linear Equations in One Variable A linear equation in one variable is an equation that can be written in the form ax + b = c where a, b, and c are real numbers and a 0. 3 x + 5 = 25 The expressions are called the sides of the equation. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 3

Solutions The solution of a linear equation is the value or values of the variable that make the equation a true statement. The set of all solutions of an equation is called the solution set. The solution satisfies the equation. Example: Determine if x = – 1 is a solution to the equation. – 3(x – 3) = – 4 x + 3 – 5 x – 3[(– 1) – 3] = – 4(– 1) + 3 – 5(– 1) – 3(– 4) = 4 + 3 + 5 True. x = – 1 is a solution 12 = 12 Copyright © 2014, 2010, 2007 Pearson Education, Inc. 4

Solving Equations Linear equations are solved by writing a series of steps that result in the equation x = a number One method for solving equations is to write a series of equivalent equations. Two or more equations that have precisely the same solutions are called equivalent equations. 3+5=8 1+7=2+6 Copyright © 2014, 2010, 2007 Pearson Education, Inc. 5

Addition Property The Addition Property of Equality states that for real numbers a, b, and c, if a = b, then a + c = b + c. 6 + y = 11 We need to find the value of y. 6 + y + ( 6) = 11 + ( 6) y=5 Solution to the equation. Adding ( 6) to both sides of the equation will maintain the balance of the equation. 6 6 Left side Right side Copyright © 2014, 2010, 2007 Pearson Education, Inc. 6

Using the Addition Property Example: Solve the linear equation x 9 = 22. Step 1: Isolate the variable x on the left side of the equation. x 9 + 9 = 22 + 9 Add 9 to both sides of the equation. Step 2: Simplify the left and right sides of the equation. x = 31 Apply the Additive Inverse Property. Step 3: Check to verify the solution. x 9 = 22 31 9 = 22 22 = 22 Copyright © 2014, 2010, 2007 Pearson Education, Inc. 7

Multiplication Property The Multiplication Property of Equality states that for real numbers a, b, and c, where c 0, if a = b, then ac = bc. We need to find the value of x. x = 28 Solution to the equation. Multiplying both sides of the equation by will maintain the balance of the equation. × 7 Left side Right side Copyright © 2014, 2010, 2007 Pearson Education, Inc. 8

Using the Multiplication Property Example: Solve the linear equation 3 x 3 = 81 Step 1: Get the coefficient of the variable x to be 1. (3 x) = (81) Multiply each side of the equation by Step 2: Simplify the left and right sides of the equation. x = 27 Apply the Multiplicative Inverse Property. Step 3: Check to verify the solution. 3 x = 81 3(27 3( ) = 81 81 = 81 Copyright © 2014, 2010, 2007 Pearson Education, Inc. 9