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Multi-Step Equations with Like Terms Andrew Gloag Eve Rawley Anne Gloag Say Thanks to the Authors Click http://www.ck12.org/saythanks (No sign in required) To access a customizable version of this book, as well as other interactive content, visit www.ck12.org CK-12 Foundation is a non-profit organization with a mission to reduce the cost of textbook materials for the K-12 market both in the U.S. and worldwide. Using an open-content, web-based collaborative model termed the FlexBook®, CK-12 intends to pioneer the generation and distribution of high-quality educational content that will serve both as core text as well as provide an adaptive environment for learning, powered through the FlexBook Platform®. 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Printed: October 25, 2013 AUTHORS Andrew Gloag Eve Rawley Anne Gloag www.ck12.org C ONCEPT Concept 1. Multi-Step Equations with Like Terms 1 Multi-Step Equations with Like Terms Here you’ll learn how to add and subtract like terms to simplify two-step equations and solve for their unknown. What if you had an equation in which the same variable appeared twice, like 2(x − 4) + 4x = −23? How could you simplify the equation so that the variable appears only once in order to solve for it? After completing this Concept, you’ll be able to combine like terms to solve two-step equations like this one. Watch This MEDIA Click image to the left for more content. CK-12 Foundation: 0305S Combining Like Terms in Two-Step Equations Guidance When we look at a linear equation we see two kinds of terms: those that contain the unknown variable, and those that don’t. When we look at an equation that has an x on both sides, we know that in order to solve it, we need to get all the x−terms on one side of the equation. This is called combining like terms. The terms with an x in them are like terms because they contain the same variable (or, as you will see in later chapters, the same combination of variables). TABLE 1.1: Like Terms x 4x, 10x, −3.5x, and 12 3y, 0.000001y, and y xy, 6xy, and 2.39xy Unlike Terms 3x and 3y 4xy and 4x 0.5x and 0.5 Example A To add or subtract like terms, we can use the Distributive Property of Multiplication. 3x + 4x = (3 + 4)x = 7x 0.03xy − 0.01xy = (0.03 − 0.01)xy = 0.02xy −y + 16y + 5y = (−1 + 16 + 5)y = 10y 5z + 2z − 7z = (5 + 2 − 7)z = 0z = 0 To solve an equation with two or more like terms, we need to combine the terms first. 1 www.ck12.org Example B Solve (x + 5) − (2x − 3) = 6. There are two like terms: the x and the −2x (don’t forget that the negative sign applies to everything in the parentheses). So we need to get those terms together. The associative and distributive properties let us rewrite the equation as x + 5 − 2x + 3 = 6, and then the commutative property lets us switch around the terms to get x − 2x + 5 + 3 = 6, or (x − 2x) + (5 + 3) = 6. (x − 2x) is the same as (1 − 2)x, or −x, so our equation becomes −x + 8 = 6 Subtracting 8 from both sides gives us −x = −2. And finally, multiplying both sides by -1 gives us x = 2. Example C Solve 2x − 3x = 6. This problem requires us to deal with fractions. We need to write all the terms on the left over a common denominator of six. 3x 2x − =6 6 6 Then we subtract the fractions to get x 6 = 6. Finally we multiply both sides by 6 to get x = 36. Watch this video for help with the Examples above. MEDIA Click image to the left for more content. CK-12 Foundation: Combing Like Terms in 2 Step Equations Vocabulary • Terms with the same variable in them (or no variable in them) are like terms. Combine like terms (adding or subtracting them from each other) to simplify the expression and solve for the unknown. Guided Practice Solve 2x 5 − 3x 2 = 11. This problem requires us to deal with fractions. We need to write all the terms on the left over a common denominator of ten. 4x 15x − = 11 10 10 2 www.ck12.org Concept 1. Multi-Step Equations with Like Terms Then we subtract the fractions to get − 11x 10 = 11. 10 : Finally we multiply both sides by − 11 10 10 − 11x 10 · − 11 = 11 · − 11 to get x = −10. Practice Solve the following equations for the unknown variable. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 1.3x − 0.7x = 12 −10a − 2(a + 5) = 14 5(2y − 3y) = −20 2 1 14 3 x − 5 x = 15 5x − (3x + 2) = 1 5 s − 3s 8 = 6 10(y + 5y) = 10 2.3x + 2(0.75x − 3.5) = 7.5 3(x + 2) + 5(2 − x) = −32 6x + 2(5x − 2) = 12 3