Name Date Class. When solving multi-step equations, first combine like terms on each side if possible. Then use inverse operations.

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1 x-x 1-x 1-4 Solving Two-Step and Multi-Step Equations When solving multi-step equations, first combine like terms on each side if possible. Then use inverse operations. 4x 3 15 Operations x is multiplied by 4. Then 3 is subtracted. Solve using Inverse Operations Add 3 to both sides. Then divide both sides by 4. x x is divided by 3. Then 2 is added. Add 2 to both sides. Then multiply both sides by 3. The order of the inverse operations is the order of operations in reverse. Solve 2x 7 3x 13. Check: 2x 3x 7 13 Group like terms together. 2x 7 3x 13 5x 7 13 Add like terms. 2(4) 7 3(4)? 13 5x 7 13 x is multiplied by 5. Then 7 is subtracted. 7 7 Add 7 to both sides ? 13 5x 20 13? 13 5x x 4 Divide both sides by 5. Solve each equation. Check your answers. 1. 3x b y 4 2y (x 2)

2 x-x 1-x 1-4 Solving Two-Step and Multi-Step Equations continued A two-step equation with fractions can be simplified by multiplying each side by the LCD. This will clear the fractions. Solve x Check: x x x (12)2 Multiply both sides by the LCD x x (2) 4 3 3x 8 24 x is multiplied by 3. 8 is added. 8 8 Add 8 to both sides. 3x 16 3x x ? ? ? 2 6 3? 2 Divide both sides by 3. 2? 2 Solve each equation. Check your answers. 5. x w a

3 Practice C 1. r 3 2. w y 4 4. f 5 5. p r 7 7. y h m v b n x , 57 cookies Reading Strategies 1. like terms 2. subtraction 3. Subtract 3 from both sides, then multiply by n 5 5. d 7 6. j 2 x SOLVING EQUATIONS WITH VARIABLES ON BOTH SIDES Practice A 1. 2a; 3; 10; r; 9; b; 30; 5b; 3b; c all real numbers 6. no solution 7. a. 3 hours 8. a. 2 hours b. 75 F b. $ Practice B 1. d n 1 3. p 4 4. t x r 2 Challenge 1. 4 inches 2. 8 inches 3. 1 inch 4. 9 inches inches 6. 7 inches 7. 3 inches 8. 7 inches 9. 6 inches inches inches inches sides sides sides sides Problem Solving d 55.79; 16 DVDs p 1; 2.9 million 3. 9 years old weeks 5. D 6. F 7. B 7. y all real numbers 9. m no solution 11. b r a. 8 years 14. a. 6 months b. $52,000 b. 128 stamps Practice C 1. x 2 2. a 3 3. c 1 4. y 3 5. d t 2 7. m 9 8. all real numbers 9. no solution 10. all real numbers shirts, $ weeks, $85 1. Possible answers: add 3x to each side, add 7x to each side 2. Possible answers: add 4x to each side, add 10x to each side A5

4 x-x 1-x 1-1 Variables and Expressions To translate words into algebraic expressions, find words like these that tell you the operation. add subtract multiply divide sum difference product quotient more less times split increased decreased per ratio Kenny owns v video games. Stan owns 7 more video games than Kenny. Write an expression for the number of video games Stan owns. v represents the number of video games Kenny owns. v 7 Think: The word more indicates addition. Order does not matter for addition. The expression 7 v is also correct. Jenny is 12 years younger than Candy. Write an expression for Jenny s age if Candy is c years old. c represents Candy s age. The word younger means less, which indicates subtraction. c 12 Think: Candy is older, so subtract 12 from her age. Order does matter for subtraction. The expression 12 c is incorrect. 1. Jared can type 35 words per minute. Write an expression for the number of words he can type in m minutes. 2. Mr. O Brien s commute to work is 0.5 hour less than Miss Santos s commute. Write an expression for the length of Mr. O Brien s commute if Miss Santos s commute is h hours. 3. Mrs. Knighten bought a box of c cookies and split them evenly between the 25 students in her classroom. Write an expression for the number of cookies each student received. 4. Enrique collected 152 recyclable bottles, and Latasha collected b recyclable bottles. Write an expression for the number of bottles they collected altogether. 5. Tammy s current rent is r dollars. Next month it will be reduced by $50. Write an expression for next month s rent in dollars. 1-6

5 x-x 1-x 1-1 The value of Variables and Expressions continued 9 depends on what number is placed in the box. Evaluate 9 when 20 is placed in the box In algebra, variables are used instead of boxes. Evaluate x 7 for x 28. x Sometimes, the expression has more than one variable. Evaluate x y for x 6 and y 2. x y Evaluate 5 when each number is placed in the box Evaluate each expression for x 4, y 6, and z x y z Evaluate each expression for x 2, y 18, and z x z 13. y x 14. y z 15. y x 16. xy 17. z x 1-7

6 Answer Key For Equations x VARIABLES AND EXPRESSIONS Practice A 1. the sum of a and times x 3. y less than 5 4. the quotient of n and increased by t 6. the product of 3 and s 7. c m a. j 4 b.11 years old; 16 years old; 54 years old Practice B 1. the difference of 15 and b; b less than the quotient of x and 16; x divided by the sum of x and 9; 9 more than x 4. the product of 2 and t; 2 times t 5. the difference of z and 7; 7 less than z 6. the product of 4 and y; 4 times y 7. g m a. d 20 b. 40 dollars; 80 dollars; 95 dollars Practice C 1. the sum of k and 2.5; 2.5 more than k 2. the product of 5 and n; 5 times n 3. the quotient of b and 25; b divided by the difference of 100 and x; x less than g p a. 25 2g b. 23 tokens; 17 tokens; 5 tokens; 1 token 11. n 8, Possible answer: Tammy is 8 years younger than Nate, who is n years old 12. 4x, Possible answer: Bo had 4 times as many votes as Scott, who had x votes 1. 35m 2. h c b 5. r Challenge (m) Expression for Plan A Cost of Plan A (200) $ (300) $ (400) $ (500) $265 Expression for Plan B Cost of Plan B (100) $ (200) $ (300) $ (400) $ (500) $ Plan A 3. Plan B A1

7 x-x 1-x 1-5 Solving Equations with Variables on Both Sides Variables must be collected on the same side of the equation before the equation can be solved. Solve 10x 2x 16. Check: 10x 2x 16 10x 2x 16 2x 2x Add 2x to both sides. 10(2)? 2(2) 16 8x 16 20? x x 2 Solve 3x 5(x 2). Divide both sides by 8. 20? 20 Check: 3x 5x 10 Distribute. 3x 5(x 2) 5x 5x Add 5x to both sides. 3(5)? 5(5 2) 2x 10 15? 5(3) 2x x 5 Divide both sides by 2. 15? 15 Write the first step you would take to solve each equation. 1. 3x 2 7x 2. 4x 6 10x 3. 15x 7 3x Solve each equation. Check your answers. 4. 4x 2 5(x 10) y 3 4y (t 7) 2 6t 2 2t 1-38

8 x-x 1-x 1-5 Solving Equations with Variables on Both Sides continued Some equations have infinitely many solutions. These equations are true for all values of the variable. Some equations have no solutions. There is no value of the variable that will make the equation true. Solve 3x 9 4x 9 7x. Check any value of x: 3x 9 3x 9 Combine like terms. Try x 4. 3x 3x Add 3x to each side. 3x 9 4x 9 7x 9 9 True statement. 3(4) 9? 4(4) 9 7(4) The solution is the set of all real numbers. 12 9? ? 3 Solve 2x 6 3x 5x 10. Check any value of x: 2x 6 3x 5x 10 Try x 1. 5x 6 5x 10 Combine like terms. 2x 6 3x 5x 10 5x 5x Add 5x to each side. 2(1) 6 3(1)? 5(1) False statement ? 5 10 There is no solution. 11? 5 Solve each equation. 7. x 2 x x 8 2x g 3g x 1 4x x (f 3) 4f 6 6f 12. 3x 7 2x 4x

9 Practice C 1. r 3 2. w y 4 4. f 5 5. p r 7 7. y h m v b n x , 57 cookies Reading Strategies 1. like terms 2. subtraction 3. Subtract 3 from both sides, then multiply by n 5 5. d 7 6. j 2 x SOLVING EQUATIONS WITH VARIABLES ON BOTH SIDES Practice A 1. 2a; 3; 10; r; 9; b; 30; 5b; 3b; c all real numbers 6. no solution 7. a. 3 hours 8. a. 2 hours b. 75 F b. $ Practice B 1. d n 1 3. p 4 4. t x r 2 Challenge 1. 4 inches 2. 8 inches 3. 1 inch 4. 9 inches inches 6. 7 inches 7. 3 inches 8. 7 inches 9. 6 inches inches inches inches sides sides sides sides Problem Solving d 55.79; 16 DVDs p 1; 2.9 million 3. 9 years old weeks 5. D 6. F 7. B 7. y all real numbers 9. m no solution 11. b r a. 8 years 14. a. 6 months b. $52,000 b. 128 stamps Practice C 1. x 2 2. a 3 3. c 1 4. y 3 5. d t 2 7. m 9 8. all real numbers 9. no solution 10. all real numbers shirts, $ weeks, $85 1. Possible answers: add 3x to each side, add 7x to each side 2. Possible answers: add 4x to each side, add 10x to each side A5

10 3. Possible answers: add 3x to each side, add 15x to each side no solution all real numbers 10. no solution 11. all real numbers Challenge 1. x x x 4 4. x x (6) 0 or 2x x 18 0 or 9x (18) x 27 0 or 5x (27) x (21) 0 or 3x a. x b a b. After writing the equation in the form ax b 0, substitute the values of a and b into the formula found in part a. 10. x x x x a. infinitely many solutions b. no solution Problem Solving feet 2. 7 days hours months 5. B 6. G 7. C Reading Strategies 1. Look for an opportunity to use the Distributive Property. 2. Combine any like terms. 3. Collect the variable terms on one side of the equal sign. 4. p 6 5. x 1 6. t 2 x SOLVING FOR A VARIABLE Practice A 1. C K f 1 T 3. y 5 x 5. m 7. w 9 v 9. a. t d r p 7n 3 4. s r 4t 6. j 6 k h 8. a bc 3 b. 3 b. 12 Practice B 1. r 2 C 3. c 4 d 5. p 7. b a c 9. a. p q 5r 2 c a. F 2 E V 2. m y b x 4. n 8 6m 6. x 8. j 10 z y h 4 k 10. a. b 2A h b. 17 b. 32 mm Practice C 1. w 3. f 5. x P 2l 2 g 3 z 7y 3 2. v f at v i 4. a 12 5b 6. h 7. r s(t 9) 8. n 9. a. a F m 10. a. t jk g 5 m 3 p I Pr b. 19 m/s 2 b. 8 years A6

11 x-x 1-6 Solving for a Variable Solving for a variable in a formula can make it easier to use that formula. The process is similar to that of solving multi-step equations. Find the operations being performed on the variable you are solving for, and then use inverse operations. Operations Solve using Inverse Operations A lw Solve for w. w is multiplied by l. Divide both sides by l. P 2l 2w Solve for w. w is multiplied by 2. Then 2l is added. Add 2l to both sides. Then divide both sides by 2. The formula A 1 bh relates the area A of a triangle 2 to its base b and height h. Solve the formula for b. A 1 2 bh b is multiplied by 1 2. The order of the inverse operations is the order of operations in reverse. 2 2 A bh Multiply both sides by A bh b is multiplied by h. 2A h 2A h bh h b Simplify. Divide both sides by h. Solve for the indicated variable. 1. P 4s for s 2. a b c 180 for b 3. P KT V for K The formula V 1 lwh relates the volume of a square pyramid 3 to its base length l, base width w, and height h. 4. Solve the formula for w. 5. A square pyramid has a volume of 560 in 3, a base length of 10 in., and a height of 14 in. What is its base width? 1-46

12 x-x 1-6 Solving for a Variable continued Any equation with two or more variables can be solved for any given variable. Solve x y z 10 x y z 10 for y. y z is divided by (x) 10 y z Multiply both sides by x y z z is subtracted from y. Add z to both sides. z z 10x z y Solve a b c d for c. a b c d b b Add b to each side. a b c d c d(a b) d d Multiply both sides by d. d(a b) c Simplify. State the first inverse operation to perform when solving for the indicated variable. 6. y x z; for z 7. f g 2 h; for g 8. t 3r s ; for s 5 Solve for the indicated variable. 9. 3ab c; for a 10. y x z m 3 ; for z n p; for m 1-47

13 1. s 4 P 3. K VP T in. 6. add x to both sides 7. multiply both sides by 2 8. add 3r to both sides 9. a 3 c b 11. m pn 3 Challenge 2. b 180 a c 4. w 3V lh 10. z 3(y x) 1. l P t 5 years 4. r or 3.2% 5. $ years 7. $ or 3% 9. $ years or 2.7% Problem Solving 1. r d t m/s m/s m/s 5. B 6. F 7. D 8. J Reading Strategies 1. Possible answer: 3x 2y 9 2. The equation contains only one variable, n. 3. Yes, because it has two or more variables. 4. Divide both sides by r. 5. t b a. h V Iw b. 3 cm 1-7 x-7 SOLVING ABSOLUTE-VALUE EQUATIONS Practice A 1. 3; 2; 2; ; 7; 4; 4; 4; 4; 11; ; 6; 6; 5; 7 4. {8, 8} 5. {14, 14} 6. {9, 9} 7. {17, 17} 8. {11, 7} 9. {1, 11} 10. {5, 5} 11. {7, 3} 12. {11, 9} 13. x miles per gallon; 26 miles per gallon Practice B 1. {12, 12} , {10, 10} 4. {9, 9} 5. {8, 8} 6. {13, 7} 7. {1, 3} 8. {2, 8} 9. {14, 10} 10. {3, 3} 11. {1} 12. {3} 13. two 14. one 15. none 16. x ; 71.5 Practice C , {0} 3. {10.5, 10.5} {7, 7} 6. {11} , A7

14 x-x 1-7 Solving Absolute-Value Equations There are three steps in solving an absolute-value equation. First use inverse operations to isolate the absolute-value expression. Then rewrite the equation as two cases that do not involve absolute values. Finally, solve these new equations. Solve x 3 4 = 8. Step 1: Isolate the absolute-value expression. x 3 4 = Subtract 4 from both sides. x 3 = 4 Step 2: Rewrite the equation as two cases. x 3 = 4 Case 1 Case 2 Step 3: x 3 = 4 x 3 = 4 Solve Add 3 to both sides. x = 1 x = 7 The solution are 1 and 7. Solve each equation. 1. x 2 3 = 5 2. x 7 2 = x 5 = x 1 =

15 x-x 1-7 Solving Absolute-Value Equations continued Some absolute-value equations have two solutions. Others have one solution or no solution. To decide how many solutions there are, first isolate the absolute-value expression. Original Equation Simplified Equation Solutions x 5 7 x x 2 x x x 5 0 x x x 7 3 x 2 has two solutions, x 2 and x 2. The solutions are 2 and 2. x 5 0 means x 5 0, so there is one solution x 5. The solution is 5. x 7 3 has no solutions because an absolute-value expression is never negative. There is no solution. Solve 2x 1 3 = 7. 2x 1 3 = Add 3 to both sides. 2x 1 = 4 Absolute value cannot be negative. The equation has no solution. Solve each equation x 2 = 8 6. x 1 5 = x 3 = x 10 =

16 9. {6.6, 8.6} 10. {8, 3} x ; m; m 14. x 5 1.5; 6.5C; 3.5C 1. {6, 10} 2. {15, 1} 3. {0, 10} 4. {3, 3} 5. {2} {10} Challenge 1. (3) 2 5 5; yes; possible answer: x x x x x x x x x Problem Solving 1. x ; cm; cm 2. x 53 = 0.021; m; m 3. 1 and cm 5. B 6. G 7. C Reading Strategies 1. one 2. two 3. none 4. one 5. none x two RATES, RATIOS, AND PROPORTIONS Practice A ft/s 3. $1.05/lb pages/min 5. y 4 6. x m 2 8. t b x in mi Practice B $0.49/lb cars/min ft/s 5. y 5 6. x m t b x in cm Practice C lb/book 3. $7.90/h ft/s 5. x b s y 2 9. x y ft in m 1km ; 3,391,100 ; 40, c/min 6. x k 3 8. a y 30 A8

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