Chapter 8 - XYZ Custom Plus

500Chapter 8 Linear Equations and InequalitiesSometimes it is necessary to simplify each side of an equation before using theaddition property of equality. The reason we simplify both sides first is that wewant as few terms as possible on each side of the equation before we use theaddition property of equality. The following examples illustrate this procedure.8. Solve 5(3a − 4) − 14a = 25.Example 8Solve 4(2a − 3) − 7a = 2 − 5.Solution We must begin by applying the distributive property to separateterms on the left side of the equation. Following that, we combine similar termsand then apply the addition property of equality.NoteAgain, we placea question markover the equal signbecause we don’t know yetwhether the expressions on theleft and right side of the equal signwill be equal.4(2a − 3) − 7a = 2 − 5 Original equation8a − 12 − 7a = 2 − 5 Distributive propertya − 12 = −3 Simplify each sidea − 12 + 12 = −3 + 12 Add 12 to each sidea = 9AdditionTo check our solution, we replace a with 9 in the original equation.4(2 ⋅ 9 − 3) − 7 ⋅ 9 ≟ 2 − 54(15) − 63 ≟ −360 − 63 ≟ −3−3 = −3 A true statement9. Solve 5x − 4 = 4x + 6.NoteIn my experienceteaching algebra,I find that studentsmake fewer mistakes if they thinkin terms of addition rather thansubtraction. So, you are probablybetter off if you continue to usethe addition property just the waywe have used it in the examples inthis section. But, if you are curiousas to whether you can subtract thesame number from both sides ofan equation, the answer is yes.Example 9Solve 3x − 5 = 2x + 7.Solution We can solve this equation in two steps. First, we add −2x to bothsides of the equation. When this has been done, x appears on the left side only.Second, we add 5 to both sides:3x + (−2x) − 5 = 2x + (−2x) + 7 Add −2x to both sidesx − 5 = 7Simplify each sidex − 5 + 5 = 7 + 5Add 5 to both sidesx = 12Simplify each sideA Note on SubtractionAlthough the addition property of equality is stated for addition only, we cansubtract the same number from both sides of an equation as well. Becausesubtraction is defined as addition of the opposite, subtracting the same quantityfrom both sides of an equation does not change the solution.x + 2 = 12 Original equationx + 2 − 2 = 12 − 2 Subtract 2 from each sidex = 10 SubtractionGetting Ready for ClassAfter reading through the preceding section, respond in your ownwords and in complete sentences.1. What is a solution to an equation?2. What are equivalent equations?3. Explain in words the addition property of equality.Answers8. 45 9. 104. How do you check a solution to an equation?