Name Reteaching 2-1 Understanding Equations A solution to an equation is a value that makes the equation true. An equation is true if both sides are equal. To find out if a given value is a solution to an equation, substitute the value for the variable. If both sides of the equation are equal, the value is a solution to the equation. Use this method to solve the following problem. Kate is going to babysit for 5 hours. She needs to make Possible Babysitting Rates exactly $20.00 to buy a concert ticket. How much does Kate need to make each hour in order to buy the concert $3.00 per hour ticket? $4.00 per hour The equation for this situation is 5x 5 $20.00, where x is $5.00 per hour the amount she makes per hour. $6.00 per hour To find the solution to 5x 5 $20.00, substitute the different rates for x. Try x 5 $3.00: 5 3 $3.00 5 $15.00 Not a solution Try x 5 $4.00: 5 3 $4.00 5 $20.00 Solution Try x 5 $5.00: 5 3 $5.00 5 $25.00 Not a solution Try x 5 $6.00: 5 3 $6.00 5 $30.00 Not a solution Since the solution is $4.00, Kate needs to make $4.00 per hour in order to earn enough money to buy the concert ticket. Substitute the different values of the variable to find the solution to each equation. 1. 5 1 x 5 22 x 5 7, 12, 17, 20 17 3. 27 4 p 5 9 p 5 3, 4, 6, 7 3 2. n 2 9 5 33 n 5 14, 24, 32, 42 42 4. 81 5 9y y 5 2, 5, 8, 9 9 Tell if each equation is true or false for k 5 4. 5. 32 4 k 5 8 True 6. k 5 31 2 27 True 7. 43.6 2 k = 38.6 False Tell which value of the variable is the solution to the equation. 8. 23.7 5 41.1 2 t t 5 17.4, 18.4, 27.4, 64.8 17.4 9. d 1 19.6 5 34.5 d 5 13.1, 13.9, 14.9, 22.9 14.9 10. Reasoning Ada is putting autographed baseballs in a display case. The display case can hold 28 baseballs, and there are already 17 baseballs in the case. Ada thinks that she can display 9 more baseballs and the display case will be full. How can you use the equation 17 1 b 5 28 to check if she is right? Sample answer: Substitute 9 for b to see if the equation is true. 17 1 9 5 26, so 9 is not a solution. Ada is incorrect. R 2•1 Copyright © Pearson Education, Inc., or its affiliates. All Rights Reserved. 6 Name Practice 2-1 Understanding Equations Substitute the different values of the variable to find the solution to each equation. 1. 27 2 c 5 18 c 5 9, 11, 35, 45 9 3. 8 3 s 5 96 s 5 9, 12, 13, 14 12 2. q 2 19 5 12 q 5 7, 21, 29, 31 31 4. 56 5 7f f 5 6, 7, 8, 9 8 Tell if each equation is true or false for w 5 2.1. 5. 28.4 2 w 5 25.3 False 6. w 5 39.2 2 37.1 True Tell which value of the variable is the solution to the equation. 7. t 1 $13.38 5 $19.00 t 5 $5.62, $5.72, $6.62, $7.72 $5.62 8. 19.7 5 41.1 2 g g 5 21.4, 22.4, 30.4, 31.4 21.4 9. 7.7 1 r 5 8.5 r 5 0.2, 0.6, 0.8, 1.2 0.8 10. Writing to Explain Lou set up 6 tables for a party. 42 people are coming to the party. Lou is planning to seat 7 people at each table. Use the equation 42 4 p 5 6 to explain whether Lou’s plan will work. Sample answer: Substitute 7 for p in the equation to see if 7 people can be seated at each table. 42 4 7 5 6, so Lou’s plan will work. 11. Reasoning 117 students and teachers participated in a fundraiser. 96 students participated. Did 11, 19, 21, or 29 teachers participate? Use the equation t 1 96 5 117 to justify your answer. 21 teachers participated. Sample answer: Substitute each number of teachers for t. 21 1 96 5 117, so 21 teachers participated. 12. Geometry Jerry built a table with a square top. The perimeter of the _1 tabletop is 18 feet. He knows that each side of the table is either 3, 3​ 2 , _1 4, or 4​ 2 feet long. Use the equation 18 5 4s to help him find which is the length of each side of the tabletop. _1 4​ 2 feet. Sample answer: Substitute each side _1 _1 length for s. 18 5 4 3 4​ 2 , so each side is 4​ 2 feet long. P 2•1 Copyright © Pearson Education, Inc., or its affiliates. All Rights Reserved. 6 Name Enrichment 2-1 Use Your Head! Calculate the answers to each problem using mental math. Mental Math 1. Juanita’s bakery ordered 48 lb of white flour, 32 lb of wheat flour, 42 lb of butter, and 38 lb of margarine. What was the total number of pounds the bakery ordered? 48 32 42 38 (48 38) (32 42) (50 40) 4 (30 40) + 4 (90) 4 (70) 4 160 lb 2. The bakery usually orders 62 lb of walnuts each week. At the end of this week, there were 14 lb left over. How many pounds of walnuts should be ordered for next week? 62 14 (62 12) 2 50 2 48 lb of walnuts 3. If the bakers bake 100 loaves of white bread every day, how many loaves do they bake every 4 weeks? 100 7 4 (7 4) 100 28 100 2,800 loaves of white bread 4. The bakers at Juanita’s bake 12 dozen potato rolls, 15 dozen crescent rolls, and 10 dozen kaiser rolls each day. What is the total number of rolls baked each day? (12 12) (15 12) (10 12) 12(12 15 10) 12(37) 10(37) 2(37) 370 74 444 rolls 5. A new baker at Juanita’s ordered 240 extra pounds of raisins. _1 If 4 lb of raisins are used for every batch of raisin muffins, and there are 10 muffins in each batch, how many muffins will the baker make before the extra pounds of raisins are used? 240 4 960 batches 10 muffins 9,600 muffins E 2•1 Copyright © Pearson Education, Inc., or its affiliates. All Rights Reserved. 6 Name Reteaching 2-2 Properties of Equality To keep an equation balanced, you must do the same thing to each side. Balanced Equation Unbalanced Equation 1 1 1 1 Add 2 Add 2 Add 2 Add 4 3 3 3 5 The scale is balanced because both sides have The equation is not balanced. 3 does not equal the same value. We added the same amount to 5. We did not add the same amount to both each side of the equation. sides of the equation. Use the Properties of Equality to balance equations. Add the same number to each side. 3c 5 12, so 3c 1 5 5 12 1 5 Subtract the same number from each side. 3c 5 12, so 3c 2 3 5 12 2 3 Multiply each side by the same number. 3c 5 12, so 3c 3 2 5 12 3 2 Divide each side by the same number. 3c 5 12, so 3c 4 4 5 12 4 4 Evaluate the equations. 1. If 16 1 5 5 21, does 16 1 5 2 4 5 21 2 4? Why or why not? Yes, because the same number, 4, is subtracted from each side. 2. If 3p 5 27, does 3p 3 2 5 27 3 3? Why or why not? No, because each side is multiplied by a different number. 3. If 4s 2 6 5 18, does (4s 2 6) 4 2 5 18 4 2? Why or why not? Yes, because each side is divided by the same number, 2. 4. Reasoning A pan balance shows x 1 2 5 10. If you add 5 units to one side, can you balance the scale by adding x units to the other side? Explain. No, to keep the scale and equation balanced, you have to add the same number of units, 5, to each side. R 2•2 Copyright © Pearson Education, Inc., or its affiliates. All Rights Reserved. 6 Name Practice 2-2 Properties of Equality 1. If 16 1 4 5 20, does 16 1 4 2 4 5 20 − 4? Why or why not? Yes, the same number is subtracted from each side. 2. If 2d 4 4 5 5, does 2d 4 4 1 6 = 5 1 4? Why or why not? No, because a different number is added to each side. 3. If 12 2 8 5 4, does (12 2 8) 4 2 5 4 3 2? Explain. No, because one side is divided by 2 and the other side is multiplied by 2.