_ 4x 4 = _-8. Model. Activity. 90 Explore 2-3 Algebra Lab: Solving Multi-Step Equations
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1 Algebra Lab Solving Multi-Step Equations You can use algebra tiles to model solving multi-step equations. Virginia SOL A.4 The student will solve multistep linear and quadratic equations in two variables, including b) justifying steps used in simplifying expressions and solving equations, and d) solving multistep linear equations algebraically and graphically. Activity Use an equation model to solve 4x Step 1 Model the equation. Step Isolate the x-term x + -5 Place 4 x-tiles and positive 1-tiles on one side of the mat. Place 5 negative 1-tiles on the other side. Step Remove zero pairs. 4x Since there are positive 1-tiles with the x-tiles, add negative 1-tiles to each side to form zero pairs. Step 4 Group the tiles x -8 Group the tiles to form zero pairs and remove the zero pairs. _ 4x 4 _-8 4 x - Separate the remaining tiles into 4 equal groups to match the 4 x-tiles. Each x-tile is paired with negative 1-tiles. The resulting equation is x -. Model Use algebra tiles to solve each equation. 1. x x x x x 1 6. x x x 1 9. What would be your first step in solving 8x ? 10. What steps would you use to solve 9x ? 90 Explore - Algebra Lab: Solving Multi-Step Equations
2 Solving Multi-Step Equations Then You solved singlestep equations. (Lesson -) Now 1Solve equations involving more than one operation. Solve equations involving consecutive integers. Why? The Tour de France is the premier cycling event in the world. The map shows the 007 Tour de France course. If the length of the shortest portion of the race can be represented by k, the expression 4k + 0 is the length of the longest stage or 6 kilometers. This can be described by the equation 4k Gand Paris Chablis FRANCE Tallard New Vocabulary multi-step equation consecutive integers number theory Virginia i SOL A.4 The student will solve multistep linear and quadratic equations in two variables, including b) justifying steps used in simplifying expressions and solving equations, using field properties and axioms of equality that are valid for the set of real numbers and its subsets; d) solving multistep linear equations algebraically and graphically; and f) solving real-world problems involving equations and systems of equations. Solve Multi-Step Equations Since the above equation requires more than one 1 step to solve, it is called a multi-step equation. To solve this equation, we must undo each operation by working backward. Example 1 Solve Multi-Step Equations a. 11x x x b. _ a x _ 11x _ x Original equation Add 4 to each side. Divide each side by 11. _ a Original equation 8 8 _ ( a ) 8(5) Multiply each side by 8. a Subtract 7 from each side. a You can check your solutions by substituting the results back into the original equations. GuidedPractice 1A. a B. _ n connected.mcgraw-hill.com 91
3 Real-World Example Write and Solve a Multi-Step Equation _ SHOPPING Hiroshi is buying a pair of water skis that are on sale for of the original price. After he uses a $5 gift certificate, the total cost before taxes is $115. What was the original price of the skis? Write an equation for the problem. Then solve the equation. Real-World Link Shoppers in Shanghai, China, can pay for purchased items at a terminal that can match the buyers fingerprints with their bank accounts. Source: Shanghai Daily Words Two thirds of the price minus 5 is 115. Variable Equation _ Let p original price of the skis. p _ p Original equation _ p Add 5 to each side. _ ( p 140 p ) _ p 10 (140) Multiply each side by _. The original price of the skis was $10. GuidedPractice A. RETAIL A music store has sold _ of their hip-hop CDs, but 10 were returned. 5 Now the store has 6 hip-hop CDs. How many were there originally? B. READING Len read _ of a graphic novel over the weekend. Monday, he 4 read more pages. If he has read 0 pages, how many pages does the book have? Solve Consecutive Integer Problems Consecutive integers are integers in counting order, such as 4, 5, and 6 or n, n + 1, and n +. Counting by two will result in consecutive even integers if the starting integer n is even and consecutive odd integers if the starting integer n is odd. Concept Summary Consecutive Integers Type Words Symbols Example Consecutive Integers Consecutive Even Integers Consecutive Odd Integers Integers that come in counting order. Even integer followed by the next even integer. Odd integer followed by the next odd integer. n, n + 1, n +,, -,, 0, 1,, n, n +, n + 4,, -, 0,, 4, n, n +, n + 4,,, 1, -, 5, 9 Lesson - Solving Multi-Step Equations Number theory is the study of numbers and the relationships between them.
4 Example Solve a Consecutive Integer Problem Study Tip Representing Consecutive Integers You can use the same expressions to represent either consecutive even integers or consecutive odd integers. It is the value of n (odd or even) that differs between the two expressions. NUMBER THEORY Write an equation for the following problem. Then solve the equation and answer the problem. Find three consecutive odd integers with a sum of -51. Let n the least odd integer. Then n + the next greater odd integer, and n + 4 the greatest of the three integers. Words The sum of three consecutive odd integers is -51. Equation n + (n + ) + (n + 4) -51. n + (n + ) + (n + 4) -51 n Original equation Subtract 6 from each side. n -57 n -57 Divide each side by. n 9 n or 7 n or 5 The consecutive odd integers are 9, 7, and 5. CHECK 9, 7, and 5 are consecutive odd integers. 9 + (7) + (5) -51 GuidedPractice. Write an equation for the following problem. Then solve the equation and answer the problem. Find three consecutive integers with a sum of 1. Check Your Understanding Step-by-Step Solutions begin on page R1. Example 1 Example 1 m f _ a _ a _ x _ c NUMBER THEORY Twelve decreased by twice a number equals -4. Write an equation for this situation and then find the number. 8. BASEBALL Among the career home run leaders for Major League Baseball, Hank Aaron has 175 fewer than twice the number that Dave Winfield has. Hank Aaron hit 755 home runs. Write an equation for this situation. How many home runs did Dave Winfield hit in his career? Example Write an equation and solve each problem. 9. Find three consecutive odd integers with a sum of Find three consecutive integers with a sum of -6. connected.mcgraw-hill.com 9
5 Practice and Problem Solving Extra Practice begins on page 815. Example t n m x _ y f_ _ r _ k _ n _ 6 + z 1. 1 _ a 5. _ - w -7-6 Example FINANCIAL LITERACY The Cell+ Cellular Phone store offers the plans shown in the table. Raul chose the business plan and has budgeted $100 per month. Write an equation for this situation, and determine how many minutes per month he can use the phone and stay within budget. Plan Flat Monthly Fee Anytime Minutes Cost per Minute After Anytime Minutes personal $ $0.0 business $ $0.15 executive $ $0.10 Example Write an equation and solve each problem. 4. Fourteen less than three fourths of a number is negative eight. Find the number. 5. Seventeen is thirteen subtracted from six times a number. What is the number? 6. Find three consecutive even integers with the sum of Find three consecutive odd integers with the sum of Find four consecutive integers with the sum of Find four consecutive integers with the sum of 4. B 0. -6m n. _ b _ _ 5 _ 4 z + 1_ _ 5-4_ 9 a _ _ 5x c _ x _ 5 6 k + _ 4_ Write an equation and solve each problem. 94 Lesson - Solving Multi-Step Equations 9. -_ 7 _ 4 - _ b 40. FAMILY The ages of three brothers are consecutive integers with the sum of 96. How old are the brothers? 41. VOLCANOES Moving lava can build up and form beaches at the coast of an island. The growth of an island in a seaward direction may be modeled as 8y + centimeters, where y represents the number of years that the lava flows. An island has expanded 60 centimeters seaward. How long has the lava flowed?
6 C 4. -5x q a _ c p - (-) _ m If 7m - 5, what is the value of 11m +? 49. If 1y , what is the value of 4y - 7? 50. If -5c , what is the value of 6c - 15? 51. AMUSEMENT PARKS An amusement park offers a yearly membership of $75 that allows for free parking and admission to the park. Members can also use the water park for an additional $5 per day. Nonmembers pay $6 for parking, $15 for admission, and $9 for the water park. a. Write and solve an equation to find the number of visits it would take for the total cost to be the same for a member and a nonmember if they both use the water park at each visit. b. Make a table for the costs of members and nonmembers after, 6, 9, 1, and 15 visits to the park. c. Plot these points on a coordinate graph and describe things you notice from the graph. 5. SHOPPING At The Family Farm, you can pick your own fruits and vegetables. a. The cost of a bag of potatoes is $1.50 less than 1_ of the price of apples. Write and solve an equation to find the cost of potatoes. b. The price of each zucchini is times the price of winter squash minus $7. Write and solve an equation to find the cost of zucchini. c. Write an equation to represent the cost of a pumpkin using the cost of the blueberries. The Family Farm Fruit Price ($) Apples 6.99/bag Pumpkins 5.00 each Blueberries.99/qt Winter squash.99 each H.O.T. Problems Use Higher-Order Thinking Skills 5. OPEN ENDED Write a problem that can be modeled by the equation x Then solve the equation and explain the solution in the context of the problem. 54. REASONING Describe the steps you can use to solve _ w REASONING Determine whether each equation has a solution. Justify your answer. a. _ a a 1 b. _ 1 + b 1 - b 1 c. _ c c CHALLENGE Determine whether the following statement is sometimes, always, or never true. Explain your reasoning. The sum of three consecutive odd integers equals an even integer. 57. WRITING IN MATH Write a paragraph explaining the order of the steps that you would take to solve a multi-step equation. connected.mcgraw-hill.com 95
7 58. Which is the best estimate for the number of minutes on the calling card advertised below? A 10 min B 0 min Virginia SOL Practice $ 10 Prepaid Calling Card Only 5.4 per Minute C 50 min D 00 min 59. GRIDDED RESPONSE The scale factor for two similar triangles is :. The perimeter of the smaller triangle is 56 cm. What is the perimeter of the larger triangle in centimeters? 60. Mr. Morrison is draining his cylindrical pool. The pool has a radius of 10 feet and a standard height of 4.5 feet. If the pool water is pumped out at a constant rate of 5 gallons per minute, about how long will it take to drain the pool? (1 f t 7.5 gal) F 7.8 min G 7 h H 5.4 h J 5. h 61. STATISTICS Look at the golf scores for the five players in the table. Player Score Which of these is the range of the golf scores? A 10 C 5 B 5 D 40 A.1, A.4.b Spiral Review 6. GAS MILEAGE A midsize car with a 4-cylinder engine travels 4 miles on a gallon of gas. This is 10 miles more than a luxury car with an 8-cylinder engine travels on a gallon of gas. How many miles does a luxury car travel on a gallon of gas? (Lesson -) 6. DEER In a recent year, 186 female deer were born in Clark County. That is 9 fewer than the number of male deer born. How many male deer were born that year? (Lesson -) Translate each equation into a verbal sentence. (Lesson ) 64. f h k m 67. p 8p - r 68. _ 5 t + 1_ t 69. 1_ v _ v GEOGRAPHY The Pacific Ocean covers about 46% of Earth. If P represents the surface area of the Pacific Ocean and E represents the surface area of Earth, write an equation for this situation. (Lesson ) Find the value of n in each equation. Then name the property that is used. (Lesson 1-) n n n n Skills Review Evaluate each expression. (Lesson 1 ) ( ) 76. _ [5(1 + 1) ] 78. [8() - 4 ] + 7(4) 96 Lesson - Solving Multi-Step Equations
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