Solving Equations with Variables on Both Sides
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1 Practice A Solving Equations with Variables on Both Sides Group the terms with the variables on one side of the equal sign and simplify. Do not solve. 1. 9p 6p t 14 2t 3. 2k 18 4k 4. 6u 48 6u 5. 7m 25 2m a a Solve. Match each equation with its solution. 7. 8n 6n 14 A. n n 8 11n B. n n 8 3n C. n n n 1 D. n n 4n E. n n 5 7 n 7 F. n Members of the Campus Roller Rink pay a yearly membership fee of $50 plus $5 each time they go skating. Nonmembers pay $10 each time they go skating. How many times would both a member and a nonmember have to go skating in a year in order to pay the same amount?
2 Practice B Solving Equations with Variables on Both Sides Group the terms with the variables on one side of the equal sign and simplify. Do not solve t 6t x 32 2x 3. j 20 4j 4. 5d 40 5d 5. 9m 28 2m x x Solve. 7. 8k 6k v 3v y 10 6y a a n n d 3 10 d Members of the Lake Shawnee Club pay $40 per summer season plus $7.50 each time they rent a boat. Nonmembers pay $12.50 each time they rent a boat. How many times would both a member and a nonmember have to rent a boat in order to pay the same amount?
3 Practice C Solving Equations with Variables on Both Sides Group the terms with variables on one side of the equal sign and simplify. Do not solve t 17 8t a 42 2a a 3. 6k k 27 4k 4. 5f 40 5f m m x x 4 15 x Solve. Check each answer. 7.18r 2 12r 6r c 15 2c w 6 8w 9 2w a 6 0.6a n n d d A golf course charges nonmembers $48 per round plus a golf cart rental fee of $24 per round. A member pays a monthly fee of $384 plus a golf cart rental fee of $8 per round. How many rounds would both a member and a nonmember have to play in order to pay the same amount?
4 Review for Mastery Solving Equations with Variables on Both Sides Some equations have like terms that are on opposite sides of the equal sign. To solve these equations, first identify the like term with the lesser coefficient. 2x 25 3x 2x and 3x are like terms. Since 3 2, 3x is the term with the 2x 25 3x lesser coefficient. Move 3x to the 2x 3x 25 3x 3x opposite side of the equation by adding 5x 25 3x to each side of the equation. 5x x 5 Complete the steps to solve the equation. 1. 5x 16 9x 2. 3a a The like terms are and. The like terms are and. The like term with the lesser The like term with the lesser coefficient is. coefficient is. 5x 16 9x 3a a 5x 16 9x 3a a x 7a 1 15 x 7a 7a 14 Solve. a 3. 8d 7d p 16 4p 5. 3n 25 2n 6. 9c 6 2c k 20 2k 8. 2h 8 2 h
5 Challenge Use the Clues You can write equations using clues to solve problems. Example: The sum of two numbers is 36. The difference of the two numbers is 8. Find the numbers. Let x = one number. Let y = the other number. The sum of the two numbers is 36. x y 36 The difference of the two numbers is 8. x y 8 Solve the system of equations. x y 36 x y 8 The solution is (22, 14), so the two numbers are 22 and 14. Solve. 1. The sum of two numbers is 33. The difference of the two numbers is 5. Find the numbers. 2. The sum of two numbers is 85. The difference of the two numbers is 13. Find the numbers. 3. One number is 18 more than another number. The sum of the two numbers is 60. Find the numbers. 4. The sum of two numbers is 99. The difference of the two numbers is 9. Find the numbers. 5. The sum of two numbers is 42. One number is twice the other number. Find the numbers. 6. The sum of two numbers is 43. One number is equal to 5 less than 3 times the other number. Find the numbers.
6 Problem Solving Solving Equations with Variables on Both Sides Write the correct answer. 1. Five added to twice Erik s age is the same as 3 times his age minus 2. How old is Erik? 2. Three times the perimeter of a triangle is the same as 75 decreased by twice the perimeter. What is the perimeter of the triangle? 3. The area of a pentagon increased by 27 is the same as four times the area of the pentagon, minus 15. What is the area of the pentagon? Choose the letter for the best answer. 5. Sandy and Suzanne are planting flower pots around the school building. Sandy has planted 33 pots and is planting at the rate of 10 pots per hour. Suzanne has planted 25 pots and is planting at the rate of 14 pots per hour. In how many hours will they have planted the same number of flower pots? A 3 hr B 2.5 hr C 2 hr D 1 hr 7. Louisa used Downtown Taxi, which charges $2 for the first mile and $1.10 for each additional mile. Pietro used Uptown Cab, which charges $5 for the first mile and $0.95 for each additional mile. They paid the same amount and traveled the same distance. How far did they travel? A 25 mi B 21 mi C 20 mi D 15 mi 4. To repair body damage on a car, AutoBody charges $125, plus $18 per hour. CarCare charges $200, plus $12 per hour. Determine the number of hours for which the two body shops will cost the same. 6. The length of the sides of a square measure 2x 5. The length of a rectangle measures 2x, and the width measures x 2. For what value of x is the perimeter of the square the same as the perimeter of the rectangle? F x 2 H x 10 G x 7 J x Toni bought some beach towels on sale for $8 each. Theo bought the same number of beach towels at the full price of $12. Toni s total was $24 less than Theo s total. How many beach towels did they each buy? F 6 towels G 8 towels H 9 towels J 12 towels
7 Reading Strategies Follow a Procedure To solve equations with variables on both sides, follow these steps. 8x 3 2x 9 Step 1: Get all variable 8x 2x 3 2x 2x 9 Subtract 2x from both terms on one side of 6x 3 9 sides. the equation. Step 2: Isolate the variable 6x Add 3 to both sides. term. Use the opposite 6x 12 operation. Step 3: Use the inverse 6x operation and calculate. x 2 Answer the following questions. 1. What is the first step to solve an equation with variables on both sides? Divide both sides by What operation was performed to get the variables on one side? 3. Write the equation that results after the first step is complete. 4. What is the second step to solve the equation? 5. What operation was performed to get 3 on the opposite side? 6. Write the equation the way it looks after the second step is complete. 7. What operation was performed last?
8 Puzzles, Twisters & Teasers All Aboard! Decide whether or not each solution is correct. Circle the letter above your answer. Then use the letters to solve the riddle. 1. 6m 4m 12 m 6 W A 2. 5n 4n 32 n 4 M I 3. 4y 2y 40 y 10 V D 4. 5n 3n 26 n 13 E Q a 2a 24 a 3 J O n 2n 37 n 2 N W 7. 12h 9h 84 h 12 P T x 83 4 x 92 9 x 81 W C 9. 9t 4t 120 t 7 G A d d d 34 I B How do you define a twip? A you take a n
9 LESSON w a 20 Practice A 1. 3p t 3. 6k u m a n 6n 14 A. n n 8 11n B. n n 8 3n C. n n n 1 D. n n 4n E. n n 5 7 n 7 8 F. n times Practice B 1. 4t x 3. 5j d m x 8 7. k v 3 9. y a n d times Practice C 1. 7t a 3. 9k f 11. n rounds Review for Mastery 12. d x; 9x; 2. 3a; 4a; 5x; 4a; 5x; 5x; 4a; 4a; 4x; 7a; 16 4 ; 4x ; 4 1; 1; 4 14; 7a 7 ; 14 7 ; 3. d p 2 5. n 5 6. c 3 7. k 1 8. h 2 Challenge and and and and and and 31 Problem Solving 1. 7 years old units square units hours 5. C 6. J 7. C 8. F Reading Strategies 1. Get all the variable terms on one side of the equation. 2. subtraction 3. 6x m x 8 7. r 2 8. c 6.5
10 4.Get the other values on the opposite side of the equation. 5. addition 6. 6x division Puzzles, Twisters & Teasers 1. W 2. I 3. D 4. E 5. O 6. N 7. T 8. W 9. A 10. I W I D E O N T W A I N
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