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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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**A multi-step equation requires more than two steps to solve**

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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**Additional Example 1: Solving Equations That Contain Like Terms**

Solve. 8x x – 2 = 37 Commutative Property of Addition 8x + 3x + 6 – 2 = 37 11x + 4 = 37 Combine like terms. – 4 – 4 Since 4 is added to 11x, subtract 4 from both sides. 11x = 33 33 11 11x = Since x is multiplied by 11, divide both sides by 11. x = 3

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

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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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**Additional Example 2A: Solving Equations That Contain Fractions**

Solve. + = – 5n 4 7 4 3 4 7 4 –3 5n = 4 ( ) ( ) Multiply both sides by 4. ( ) ( ) ( ) 5n 4 7 –3 = 4 Distributive Property ( ) ( ) ( ) 5n 4 7 –3 = 4 Simplify. 5n + 7 = –3

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**Additional Example 2A Continued**

– 7 –7 Since 7 is added to 5n, subtract from both sides. 5n = –10 5n 5 –10 = Since n is multiplied by 5, divide both sides by 5 n = –2

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The least common denominator (LCD) is the smallest number that each of the denominators will divide into evenly. Remember!

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**Additional Example 2B: Solving Equations That Contain Fractions**

Solve. + – = x 2 7x 9 17 2 3 ( ) ( ) x 2 3 7x 9 17 – = 18 Multiply both sides by 18, the LCD. 18( ) + 18( ) – 18( ) = 18( ) 7x 9 x 2 17 3 Distributive Property 18( ) + 18( ) – 18( ) = 18( ) 7x 9 x 2 17 3 2 9 2 6 Simplify. 1 1 1 1 14x + 9x – 34 = 12

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**Additional Example 2B Continued**

23x – 34 = Combine like terms. Since 34 is subtracted from x, add 34 to both sides. 23x = 46 = 23x 23 46 Since x is multiplied by 23, divide t both sides by 23. x = 2

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**Distributive Property**

Check It Out! Example 2A Solve. + = – 3n 4 5 4 1 4 5 4 –1 3n = 4 ( ) ( ) Multiply both sides by 4. ( ) ( ) ( ) 3n 4 5 –1 = 4 Distributive Property ( ) ( ) ( ) 3n 4 5 –1 = 4 1 1 1 Simplify. 1 1 1 3n + 5 = –1

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**Check It Out! Example 2A Continued**

– 5 – Since 5 is added to 3n, subtract 5 from both sides. 3n = –6 3n 3 –6 = Since n is multiplied by 3, divide both sides by 3. n = –2

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**Multiply both sides by 9, the LCD. x 3 1 5x 9 13 ( ) ( ) 9 + – = 9**

Check It Out! Example 2B Solve. + – = x 3 5x 9 13 1 3 Multiply both sides by 9, the LCD. x 3 1 5x 9 13 ( ) ( ) – = 9 9( ) + 9( ) – 9( ) = 9( ) 5x 9 x 3 13 1 Distributive Property 9( ) + 9( ) – 9( ) = 9( ) 5x 9 x 3 13 1 1 3 1 3 Simplify. 1 1 1 1 5x + 3x – 13 = 3

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**Check It Out! Example 2B Continued**

8x – 13 = Combine like terms. Since 13 is subtracted from 8x, add 13 to both sides. 8x = 16 = 8x 8 16 Since x is multiplied by 8, divide t both sides by 8. x = 2

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Check It Out! Example 3 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. Penelope’s average speed is her total distance for the two days divided by the total time. Total distance Total time = average speed

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**Check It Out! Example 3 Continued**

3 + 7 = 20 m + 2m Substitute m + 2m for total distance and for total time. 10 = 20 3m Simplify. = 10(20) 10 3m Multiply both sides by 10. 3m = 200 3m 3 = Divide both sides by 3. m 66.67 Penelope rode approximately 66.7 miles.

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