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3-3 Solving Multi-Step Equations California Evaluating Algebraic Expressions Standards Extension of AF4.1 Solve two-step linear equations in one variable over the rational numbers. 3-3 Solving Multi-Step Equations Evaluating Algebraic Expressions 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. 3-3 Solving Multi-Step Equations Additional Example 1: Solving Equations That Contain Like Terms Notes Evaluating Algebraic Expressions Solve. 8x + 6 + 3x – 2 = 37 8x + 3x + 6 – 2 = 37 11x + 4 = 37 –4 11x –4 = 33 11x = 33 11 11 x=3 Commutative Property of Addition Combine like terms. Since 4 is added to 11x, subtract 4 from both sides. Since x is multiplied by 11, divide both sides by 11. 3-3 Solving Multi-Step Equations Check It Out! Example 1 Solve. Evaluating Algebraic Expressions 9x + 5 + 4x – 2 = 42 9x + 4x + 5 – 2 = 42 13x + 3 = 42 –3 13x Elbow Partners –3 = 39 13x = 39 13 13 x=3 Commutative Property of Addition Combine like terms. Since 3 is added to 13x, subtract 3 from both sides. Since x is multiplied by 13, divide both sides by 13. 3-3 Solving Multi-Step Equations Notes – Give an example Evaluating Algebraic Expressions 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. 3-3 Solving Multi-Step Equations Additional Example 2A: Solving Equations That Contain Fractions Notes Evaluating Algebraic Expressions Solve. 5n + 7= – 3 4 4 4 Multiply both sides by 4. () () () 7 = 4 –3 4(5n + 4 (4 ) (4) 4) 7 4 5n + 4 4 4 = 4 –3 4 5n + 7 = –3 Distributive Property Simplify. 3-3 Solving Multi-Step Equations Additional Example 2A Continued Notes Continued Evaluating Algebraic Expressions 5n + 7 = –3 – 7 –7 Since 7 is added to 5n, subtract 7 from both sides. 5n = –10 5n= –10 5 5 n = –2 Since n is multiplied by 5, divide both sides by 5 3-3 Solving Multi-Step Equations Give your face partner an example of this Evaluating Algebraic Expressions Remember! The least common denominator (LCD) is the smallest number that each of the denominators will divide into evenly. 3-3 Solving Multi-Step Equations Check It Out! Example 2A Solve. Evaluating 3n+ 5 = – 1 4 4 4 Algebraic Expressions ( ) ( ) 5 = 4 –1 4(3n + 4 (4 ) (4) 4) 5 = 4 –1 4(3n + 4 (4 ) (4) 4) 4 3n + 5 4 4 1 = 4 –1 4 1 1 Rally Coach A 1 1 3n + 5 = –1 1 Multiply both sides by 4. Distributive Property Simplify. 3-3 Solving Multi-Step Equations Check It Out! Example 2A Continued Evaluating Algebraic Expressions 3n + 5 = –1 – 5 –5 3n = –6 3n= –6 3 3 n = –2 Since 5 is added to 3n, subtract 5 from both sides. Since n is multiplied by 3, divide both sides by 3. 3-3 Solving Multi-Step Equations Check It Out! Example 2B Solve. Evaluating Algebraic 5x + x – 13 = 1 3 3 9 9 ( ) Expressions () Multiply both sides by 9, the LCD. () Distributive Property x – 13 = 9 1 9 5x + 3 9 9 3 () 5x 9 9 + 9 () () x 3 () () 3 5x x 9 9 +9 3 1 1 () 13 – 9 9 1 1 = 9 3 3 13 –9 9 =9 1 1 () 5x + 3x – 13 = 3 Rally Coach B 1 3 1 Simplify. 3-3 Solving Multi-Step Equations Check It Out! Example 2B Continued 8x –Evaluating 13 = 3 Combine like terms. Algebraic Expressions + 13 + 13 8x = 16 8x = 16 8 8 x=2 Since 13 is subtracted from 8x, add 13 to both sides. t Since x is multiplied by 8, divide both sides by 8. 3-3 Solving Multi-Step Equations Additional Example 3: Travel Application On Monday, DavidAlgebraic rides his bicycle m miles in Evaluating Expressions 2 hours. On Tuesday, he rides three times as far in 5 hours. If his average speed for the two days is 12 mi/h, how far did he ride on Monday? Round your answer to the nearest tenth of a mile. David’s average speed is his total distance for the two days divided by the total time. Put formula in notes Total distance = average speed Total time 3-3 Solving Multi-Step Equations Additional Example 3 Continued m + 3m = 12 Evaluating 2+5 4m = 12 7 Substitute m + 3m for total Algebraic Expressions distance and 2 + 5 for total time. Simplify. 7 4m = 7(12) Multiply both sides by 7. 7 4m = 84 4m = 84 Divide both sides by 4. 4 4 m = 21 David rode 21.0 miles.