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Evaluating Algebraic Expressions 3-3Solving Multi-Step Equations Extension of AF4.1 Solve two-step linear equations and inequalities in one variable over the rational numbers, interpret the solution or solutions in the context from which they arose, and verify the reasonableness of the results. California Standards

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Evaluating Algebraic Expressions 3-3Solving Multi-Step Equations A multi-step equation requires more than two steps to solve. To solve a multi-step equation, you may have to simplify the equation first by combining like terms.

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Evaluating Algebraic Expressions 3-3Solving Multi-Step Equations Solve. 8x x – 2 = 37 Additional Example 1: Solving Equations That Contain Like Terms 11x + 4 = 37 Combine like terms. – 4 – 4 Since 4 is added to 11x, subtract 4 from both sides. 11x = 33 x = 3 Since x is multiplied by 11, divide both sides by x 11 = 8x + 3x + 6 – 2 = 37 Commutative Property of Addition

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Evaluating Algebraic Expressions 3-3Solving Multi-Step Equations Solve. 9x x – 2 = 42 Check It Out! Example 1 13x + 3 = 42 Combine like terms. – 3 – 3 Since 3 is added to 13x, subtract 3 from both sides. 13x = 39 x = 3 Since x is multiplied by 13, divide both sides by x 13 = 9x + 4x + 5 – 2 = 42 Commutative Property of Addition

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Evaluating Algebraic Expressions 3-3Solving Multi-Step Equations If an equation contains fractions, it may help to multiply both sides of the equation by the least common denominator (LCD) to clear the fractions before you isolate the variable.

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Evaluating Algebraic Expressions 3-3Solving Multi-Step Equations Solve. + = – Additional Example 2A: Solving Equations That Contain Fractions n5n 4 Multiply both sides by –3 4 5n5n = 4 ( ) ( ) ( ) ( ) ( ) 5n – = 4 5n + 7 = –3 Distributive Property ( ) ( ) ( ) 5n – = 4 Simplify.

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Evaluating Algebraic Expressions 3-3Solving Multi-Step Equations Additional Example 2A Continued 5n + 7 = –3 – 7 –7 Since 7 is added to 5n, subtract 7 from both sides. 5n = –10 5n5n 5 –10 5 = Since n is multiplied by 5, divide both sides by 5 n = –2

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Evaluating Algebraic Expressions 3-3Solving Multi-Step Equations The least common denominator (LCD) is the smallest number that each of the denominators will divide into evenly. Remember!

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Evaluating Algebraic Expressions 3-3Solving Multi-Step Equations Solve. + – = Additional Example 2B: Solving Equations That Contain Fractions 2 3 x 2 7x7x ( ) + 18 ( ) – 18 ( ) = 18 ( ) 7x7x 9 x x + 9x – 34 = 12 ( ) ( ) x x7x – = 18 Distributive Property Multiply both sides by 18, the LCD. 18 ( ) + 18 ( ) – 18 ( ) = 18 ( ) 7x7x 9 x Simplify

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Evaluating Algebraic Expressions 3-3Solving Multi-Step Equations Additional Example 2B Continued 23x = 46 = 23x Since x is multiplied by 23, divide t both sides by 23. x = Since 34 is subtracted from 23x, add 34 to both sides. 23x – 34 = 12 Combine like terms.

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Evaluating Algebraic Expressions 3-3Solving Multi-Step Equations Solve. + = – Check It Out! Example 2A n3n 4 Multiply both sides by –1 4 3n3n = 4 ( ) ( ) ( ) ( ) ( ) 3n – = 4 3n + 5 = –1 Distributive Property ( ) ( ) ( ) 3n – = 4 Simplify

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Evaluating Algebraic Expressions 3-3Solving Multi-Step Equations Check It Out! Example 2A Continued 3n + 5 = –1 – 5 –5 Since 5 is added to 3n, subtract 5 from both sides. 3n = –6 3n3n 3 –6 3 = Since n is multiplied by 3, divide both sides by 3. n = –2

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Evaluating Algebraic Expressions 3-3Solving Multi-Step Equations Solve. + – = Check It Out! Example 2B 1 3 x 3 5x5x ( ) + 9 ( ) – 9 ( ) = 9 ( ) 5x5x 9 x x + 3x – 13 = 3 x x5x ( ) ( ) 9 + – = 9 Distributive Property Multiply both sides by 9, the LCD. 9 ( ) + 9 ( ) – 9 ( ) = 9 ( ) 5x5x 9 x Simplify

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Evaluating Algebraic Expressions 3-3Solving Multi-Step Equations 8x = 16 = 8x8x Since x is multiplied by 8, divide t both sides by 8. x = Since 13 is subtracted from 8x, add 13 to both sides. 8x – 13 = 3 Combine like terms. Check It Out! Example 2B Continued

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Evaluating Algebraic Expressions 3-3Solving Multi-Step Equations Check It Out! Example 3 Penelopes average speed is her total distance for the two days divided by the total time. average speed = Total distance Total time On Saturday, Penelope rode her scooter m miles in 3 hours. On Sunday, she rides twice as far in 7 hours. If her average speed for two days is 20 mi/h, how far did she ride on Saturday? Round your answer to the nearest tenth of a mile.

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Evaluating Algebraic Expressions 3-3Solving Multi-Step Equations Check It Out! Example 3 Continued Multiply both sides by 10. Substitute m + 2m for total distance and for total time = 20 m + 2m 10 = 20 3m Simplify. 10 = 10(20) 10 3m 3m = 200 Penelope rode approximately 66.7 miles. Divide both sides by 3. m m33m3 =

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