8 LESSON 8.1. Solving Systems of Linear Equations. Solving Systems of Linear Equations by Graphing. Solving Special Systems ESSENTIAL QUESTION

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1 Solving Sstems of Linear Equations? MODULE 8 LESSON 8.1 ESSENTIAL QUESTION Solving Sstems of Linear Equations b Graphing How can ou use sstems of equations to solve real-world problems? LESSON 8. Solving Sstems b Substitution LESSON 8.3 Solving Sstems b Elimination LESSON 8. Solving Sstems b Elimination with Multiplication LESSON 8.5 Image Credits: Kenn Ferguson/Alam Images Solving Special Sstems Real-World Video The distance contestants in a race travel over time can be modeled b a sstem of equations. Solving such a sstem can tell ou when one contestant will overtake another who has a head start, as in a boating race or marathon. Math On the Spot Animated Math Personal Math Trainer Go digital with our write-in student edition, accessible on an device. Scan with our smart phone to jump directl to the online edition, video tutor, and more. Interactivel eplore ke concepts to see how math works. Get immediate feedback and help as ou work through practice sets. 3

2 Reading Start-Up Visualize Vocabular Use the words to complete the graphic. m (, ) = m + b (- b m, 0) (0, b) Understand Vocabular Complete the sentences using the preview words. Vocabular Review Words linear equation (ecuación lineal) ordered pair (par ordenado) slope (pendiente) slope-intercept form (forma pendiente intersección) -ais (eje ) -intercept (intersección con el eje ) -ais (eje ) -intercept (intersección con el eje ) Preview Words solution of a sstem of equations (solución de un sistema de ecuaciones) sstem of equations (sistema de ecuaciones) 1. A is an ordered pair that satisfies all the equations in a sstem.. A set of two or more equations that contain two or more variables is called a. Active Reading Four-Corner Fold Before beginning the module, create a four-corner fold to help ou organize what ou learn about solving sstems of equations. Use the categories Solving b Graphing, Solving b Substitution, Solving b Elimination, and Solving b Multiplication. As ou stud this module, note similarities and differences among the four methods. You can use our four-corner fold later to stud for tests and complete assignments. Unit 3

3 Are YOU Read? Complete these eercises to review skills ou will need for this module. Simplif Algebraic Epressions Personal Math Trainer Online Assessment and Intervention EXAMPLE Simplif Group like terms. Combine like terms. Simplif a b + 3a Graph Linear Equations EXAMPLE Graph = - 1_ 3 +. Step 1: Make a table of values. =- 1_ + (, ) 3 Graph each equation. 5. = = 1_ - - O 0 = - 1_ (0) + = 3 (0, ) 3 = - 1_ (3) + = 1 3 (3, 1) Step : Plot the points. Step 3: Connect the points with a line. - - O = - O - O Module 8 5

4 Are YOU Read? (cont'd) Complete these eercises to review skills ou will need for this module. Simplif Algebraic Epressions 8. Simplif Show our work, and eplain each step. 9. Marina said that can be simplified to What was her error? Graph Linear Equations 10. Complete the table of values to determine four points that can be used to graph = 3 +. = 3 + (, ) -1 = 3(-1) + (-1, -1) 0 = 3(0) + 1 = 3(1) + = 3() Graph the equation = Is the point (-1, ) on the graph? Wh or wh not? O 6 Unit 3

5 ? L E S S O N 8.1 Solving Sstems of Linear Equations b Graphing ESSENTIAL QUESTION How can ou solve a sstem of equations b graphing? Students will solve a sstem of equations b graphing. EXPLORE ACTIVITY Investigating Sstems of Equations You have learned several was to graph a linear equation in slope-intercept form. For eample, ou can use the slope and -intercept or ou can find two points that satisf the equation and connect them with a line. A Graph the pair of equations together: { = 3 - = Slope-intercept form is = m + b, where m is the slope and b is the -intercept. 5 B Eplain how to tell whether (, -1) is a solution of the equation = 3 - without using the graph. -5 O 5 C Eplain how to tell whether (, -1) is a solution of the equation = without using the graph. -5 D E Use the graph to eplain whether (, -1) is a solution of each equation. Determine if the point of intersection is a solution of both equations. Point of intersection:, = 3 - ( ) = = 3-1 = = = The point of intersection is / is not the solution of both equations. Lesson 8.1 7

6 Math On the Spot Solving Sstems Graphicall An ordered pair (, ) is a solution of an equation in two variables if substituting the - and -values into the equation results in a true statement. A sstem of equations is a set of equations that have the same variables. An ordered pair is a solution of a sstem of equations if it is a solution of ever equation in the set. Since the graph of an equation represents all ordered pairs that are solutions of the equation, if a point lies on the graphs of two equations, the point is a solution of both equations and is, therefore, a solution of the sstem. EXAMPLE 1 M Notes Solve each sstem b graphing. { = - + A = 3 STEP 1 Start b graphing each equation. 5 STEP Find the point of intersection of the two lines. It appears to be (1, 3). Check b substitution to determine if it is a solution to both equations. -5 O 5 = - + = 3-5 3? = -(1) + 3? = 3(1) 3 = 3 3 = 3 B The solution of the sstem is (1, 3). { = 3-3 = 3( - 1) STEP 1 STEP Start b graphing each equation. Identif an ordered pairs that are solutions of both equations. The graphs of the equations are the same line. So, ever ordered pair that is a solution of one equation is also a solution of the other equation. The sstem has infinitel man solutions O 5 8 Unit 3

7 Reflect 1. A sstem of linear equations has infinitel man solutions. Does that mean an ordered pair in the coordinate plane is a solution?. Can ou show algebraicall that both equations in part B represent the same line? If so, eplain how. YOUR TURN Solve each sstem b graphing. Check b substitution. 3. { = - + = Check: O { = = 3 Check: O Personal Math Trainer Online Assessment and Intervention Lesson 8.1 9

8 Math On the Spot Solving Problems Using Sstems of Equations When using graphs to solve a sstem of equations, it is best to rewrite both equations in slope-intercept form for ease of graphing. To write an equation in slope-intercept form starting from a + b = c: a + b = c b = c - a = c - a b b = - a b + c b Subtract a from both sides. Divide both sides b b. Rearrange the equation. EXAMPLE Keisha and her friends visit the concession stand at a football game. The stand charges $ for a hot dog and $1 for a drink. The friends bu a total of 8 items for $11. Tell how man hot dogs and how man drinks the bought. STEP 1 Let represent the number of hot dogs the bought and let represent the number of drinks the bought. Write an equation representing the number of items the purchased. Number of hot dogs + Number of drinks = Total items + = 8 30 Unit 3 STEP Write an equation representing the mone spent on the items. Cost of 1 hot dog times + Cost of 1 drink times number of hot dogs number of drinks Drinks O = Total cost + 1 = 11 Write the equations in slope-intercept form. Then graph. + = 8 = 8 - = = 11 1 = 11 - = Graph the equations = and = Hot dogs Image Credits: Weronica Ankorhorn/Houghton Mifflin Harcourt

9 STEP 3 STEP Use the graph to identif the solution of the sstem of equations. Check our answer b substituting the ordered pair into both equations. Apparent solution: (3, 5) Check: + = 8 + = =? 8 (3) + 5 =? 11 8 = 8 11 = 11 The point (3, 5) is a solution of both equations. Interpret the solution in the original contet. Keisha and her friends bought 3 hot dogs and 5 drinks. Animated Math Reflect 5. Conjecture Wh do ou think the graph is limited to the first quadrant? YOUR TURN 6. During school vacation, Marquis wants to go bowling and to pla laser tag. He wants to pla 6 total games but needs to figure out how man of each he can pla if he spends eactl $0. Each game of bowling is $ and each game of laser tag is $. a. Let represent the number of games Marquis bowls and let represent the number of games of laser tag Marquis plas. Write a sstem of equations that describes the situation. Then write the equations in slope-intercept form. b. Graph the solutions of both equations. c. How man games of bowling and how man games of laser tag will Marquis pla? Games of laser tag O Games of bowling Personal Math Trainer Online Assessment and Intervention Lesson

10 Guided Practice Solve each sstem b graphing. (Eample 1) 1. { = 3 - = +. { - 3 = = O - - O Mrs. Morales wrote a test with 15 questions covering spelling and vocabular. Spelling questions () are worth 5 points and vocabular questions () are worth 10 points. The maimum number of points possible on the test is 100. (Eample )? a. Write an equation in slope-intercept form to represent the number of questions on the test. b. Write an equation in slope-intercept form to represent the total number of points on the test. c. Graph the solutions of both equations. d. Use our graph to tell how man of each question tpe are on the test. ESSENTIAL QUESTION CHECK-IN. When ou graph a sstem of linear equations, wh does the intersection of the two lines represent the solution of the sstem? Vocabular questions O Spelling questions 3 Unit 3

11 Name Class Date 8.1 Independent Practice 5. Vocabular A is a set of equations that have the same variables. 6. Eight friends started a business. The will wear either a baseball cap or a shirt imprinted with their logo while working. The want to spend eactl $36 on the shirts and caps. Shirts cost $6 each and caps cost $3 each. a. Write a sstem of equations to describe the situation. Let represent the number of shirts and let represent the number of caps. Personal Math Trainer Online Assessment and Intervention 7. Multistep The table shows the cost for bowling at two bowling alles. Shoe Rental Fee Cost per Game Bowl-o-Rama $.00 $.50 Bowling Pinz $.00 $.00 a. Write a sstem of equations, with one equation describing the cost to bowl at Bowl-o-Rama and the other describing the cost to bowl at Bowling Pinz. For each equation, let represent the number of games plaed and let represent the total cost. b. Graph the sstem. What is the solution and what does it represent? 0 Business Logo Wear b. Graph the sstem. What is the solution and what does it represent? Cost of Bowling 16 Caps 1 8 O Shirts Cost ($) O 6 8 Games Lesson

12 8. Multi-Step Jerem runs 7 miles per week and increases his distance b 1 mile each week. Ton runs 3 miles per week and increases his distance b miles each week. In how man weeks will Jerem and Ton be running the same distance? What will that distance be? 9. Critical Thinking Write a real-world situation that could be represented b the sstem of equations shown below. = + 10 { = FOCUS ON HIGHER ORDER THINKING Work Area 10. Multistep The table shows two options provided b a high-speed Internet provider. Setup Fee ($) Cost per Month ($) Option Option No setup fee $0 a. In how man months will the total cost of both options be the same? What will that cost be? b. If ou plan to cancel our Internet service after 9 months, which is the cheaper option? Eplain. 11. Draw Conclusions How man solutions does the sstem formed b - = 3 and a - a + 3a = 0 have for a nonzero number a? Eplain. 3 Unit 3

13 L E S S O N 8. Solving Sstems b Substitution Students will use substitution to solve a sstem of linear equations.? ESSENTIAL QUESTION How do ou use substitution to solve a sstem of linear equations? Solving a Linear Sstem b Substitution The substitution method is used to solve sstems of linear equations b solving an equation for one variable and then substituting the resulting epression for that variable into the other equation. The steps for this method are as follows: 1. Solve one of the equations for one of its variables. Math On the Spot. Substitute the epression from step 1 into the other equation and solve for the other variable. 3. Substitute the value from step into either original equation and solve to find the value of the variable in step 1. EXAMPLE 1 Solve the sstem of linear equations b substitution. Check our answer. { -3 + = 1 + = 8 STEP 1 Solve an equation for one variable. M Notes -3 + = 1 Select one of the equations. STEP STEP 3 = Substitute the epression for in the other equation and solve. + (3 + 1) = = 8 7 = 7 = 1 Solve for the variable. Isolate on one side. Substitute the epression for the variable. Combine like terms. Subtract 1 from each side. Divide each side b 7. Substitute the value of ou found into one of the equations and solve for the other variable,. -3 (1) + = = 1 = Substitute the value of into the first equation. Simplif. Add 3 to each side. So, (1, ) is the solution of the sstem. Lesson 8. 35

14 M Notes STEP Check the solution b graphing = 1 + = 8 -intercept: 1 3 -intercept: -intercept: 1 -intercept: O The point of intersection is (1, ). Reflect 1. Is it more efficient to solve -3 + = 1 for? Wh or wh not?. Is there another wa to solve the sstem? Personal Math Trainer Online Assessment and Intervention 3. What is another wa to check our solution? YOUR TURN Solve each sstem of linear equations b substitution. { 3 + = 11-3 = = 1 5. { + 6 = { - = = 8 36 Unit 3

15 Using a Graph to Estimate the Solution of a Sstem You can use a graph to estimate the solution of a sstem of equations before solving the sstem algebraicall. EXAMPLE Math On the Spot Solve the sstem { - = - 3 = -3. STEP 1 Sketch a graph of each equation b substituting values for and generating values of O Math Talk Mathematical Processes In Step, how can ou tell that (-5, -) is not the solution? STEP STEP 3 STEP Find the intersection of the lines. The lines appear to intersect near (-5, -). Solve the sstem algebraicall. Solve - = for. Substitute to find. - = ( + ) - 3 = -3 = = = -3 5 = -11 = The solution is (- 5, ). Use the estimate ou made using the graph to judge the reasonableness of our solution. Substitute to find. = + = + ( ) 0 - = 5 = is close to the estimate of -5, and - 11 is close to the estimate 5 5 of -, so the solution seems reasonable. Lesson 8. 37

16 YOUR TURN Personal Math Trainer Online Assessment and Intervention 7. Estimate the solution of the sstem { + = - = 6 b sketching a graph of each linear function. Then solve the sstem algebraicall. Use our estimate to judge the reasonableness of our solution. The estimated solution is. The algebraic solution is. The solution is/is not reasonable because - - O - - Solving Problems with Sstems of Equations Math On the Spot EXAMPLE 3 As part of Class Da, the eighth grade is doing a treasure hunt. Each team is given the following riddle and map. At what point is the treasure located? 10 Math Talk Mathematical Processes Where do the lines appear to intersect? How is this related to the solution? There s pirate treasure to be found. So search on the island, all around. Draw a line through A and B. Then another through C and D. Dance a jig, X marks the spot. Where the lines intersect, that s the treasure s plot! STEP 1 Give the coordinates of each point and find the slope of the line through each pair of points. A: (, 1) C: ( 1, ) B: (, 5) D: (1, ) Slope: 5 - (-1) - (-) = _ 6 = _ 3 Slope: (-1) = -8 = - -6 C 0 A -10 D B 6 38 Unit 3

17 STEP STEP 3 Write equations in slope-intercept form describing the line through points A and B and the line through points C and D. Line through A and B: Use the slope and a point to find b. 5 = ( 3 _ ) + b b = The equation is = _ 3 +. Line through C and D: Use the slope and a point to find b. = -(-1) + b b = 0 The equation is = -. Solve the sstem algebraicall. Substitute _ 3 + for in = - to find. 3_ + = - 11 = - = - 11 Substitute to find. = -( - 11) = The solution is (- 11, ). M Notes YOUR TURN 8. Ace Car Rental rents cars for dollars per da plus dollars for each mile driven. Carlos rented a car for das, drove it 160 miles, and spent $10. Vanessa rented a car for 1 da, drove it 0 miles, and spent $80. Write equations to represent Carlos s epenses and Vanessa s epenses. Then solve the sstem and tell what each number represents. Personal Math Trainer Online Assessment and Intervention Lesson 8. 39

18 Guided Practice Solve each sstem of linear equations b substitution. (Eample 1) { 3 - = = - 7 { = - + = 5 3. { + = 6. = { + = 6 - = 3 Solve each sstem. Estimate the solution first. (Eample ) 5. { 6 + = 6. - = 19 { + = = 6 Estimate Estimate Solution Solution 7. { 3 + = + 7 = 5 - = 8. { + = -1 Estimate Estimate Solution Solution 9. Adult tickets to Space Cit amusement park cost dollars. Children s tickets cost dollars. The Henson famil bought 3 adult and 1 child tickets for $163. The Garcia famil bought adult and 3 child tickets for $17. (Eample 3) a. Write equations to represent the Hensons cost and the Garcias cost. Hensons cost: Garcias cost: b. Solve the sstem.? adult ticket price: ESSENTIAL QUESTION CHECK-IN child ticket price: 10. How can ou decide which variable to solve for first when ou are solving a linear sstem b substitution? 0 Unit 3

19 Name Class Date 8. Independent Practice 11. Check for Reasonableness Zach solves the sstem { + = -3 - = 1 and finds the solution (1, -). Use a graph to eplain whether Zach s solution is reasonable. 1. Represent Real-World Problems Angelo bought apples and bananas at the fruit stand. He bought 0 pieces of fruit and spent $ Apples cost $0.50 and bananas cost $0.75 each. - Personal Math Trainer Online Assessment and Intervention - - O - a. Write a sstem of equations to model the problem. (Hint: One equation will represent the number of pieces of fruit. A second equation will represent the mone spent on the fruit.) b. Solve the sstem algebraicall. Tell how man apples and bananas Angelo bought. 13. Represent Real-World Problems A jar contains n nickels and d dimes. There is a total of 00 coins in the jar. The value of the coins is $1.00. How man nickels and how man dimes are in the jar? 1. Multistep The graph shows a triangle formed b the -ais, the line 3 - = 0, and the line + = 10. Follow these steps to find the area of the triangle. 3 - = 0 a. Find the coordinates of point A b solving the sstem { + = 10. Point A: b. Use the coordinates of point A to find the height of the triangle. height: 10 3-=0 8 6 A +=10 B O c. What is the length of the base of the triangle? base: d. What is the area of the triangle? Lesson 8. 1

20 15. Jed is graphing the design for a kite on a coordinate grid. The four vertices of the kite are at A (- _, B, C, and D. 3, _ 3 ) ( 1,- _ 3 3 ) ( 1 3, ) ( _ 3, ) One kite strut will connect points A and C. The other will connect points B and D. Find the point where the struts cross. FOCUS ON HIGHER ORDER THINKING 6-3 = Analze Relationships Consider the sstem + 3 = -8. Describe three different substitution methods that can be used to solve this sstem. Then solve the sstem. { Work Area 17. Communicate Mathematical Ideas Eplain the advantages, if an, that solving a sstem of linear equations b substitution has over solving the same sstem b graphing. 18. Persevere in Problem Solving Create a sstem of equations of the form A + B = C { that has (7, ) as its solution. Eplain how ou found the D + E = F sstem. Unit 3

21 L E S S O N 8.3 Solving Sstems b Elimination Students will solve a sstem of linear equations b adding or subtracting.? ESSENTIAL QUESTION Solving a Linear Sstem b Adding The elimination method is another method used to solve a sstem of linear equations. In this method, one variable is eliminated b adding or subtracting the two equations of the sstem to obtain a single equation in one variable. The steps for this method are as follows: 1. Add or subtract the equations to eliminate one variable.. Solve the resulting equation for the other variable. 3. Substitute the value into either original equation to find the value of the eliminated variable. How do ou solve a sstem of linear equations b adding or subtracting? Math On the Spot EXAMPLE 1 Solve the sstem of equations b adding. Check our answer. - 3 = 1 { + 3 = 6 M Notes STEP 1 Add the equations. - 3 = = 6 Write the equations so that like terms are aligned. Notice that the terms -3 and 3 are opposites. STEP = 18 3 = = 18 3 = 6 Substitute the solution into one of the original equations and solve for. + 3 = 6 Add to eliminate the variable. Simplif and solve for. Divide each side b 3. Simplif. Use the second equation = 6 3 = 0 = 0 Substitute 6 for the variable. Subtract 6 from each side. Divide each side b 3 and simplif. Lesson 8.3 3

22 STEP 3 STEP Write the solution as an ordered pair: (6, 0) Check the solution b graphing. - 3 = = 6 -intercept: 6 -intercept: 6 -intercept: - -intercept: Math Talk Mathematical Processes Is it better to check a solution b graphing or b substituting the values in the original equations? - - O - - The point of intersection is (6, 0). Reflect 1. Can this linear sstem be solved b subtracting one of the original equations from the other? Wh or wh not?. What is another wa to check our solution? Personal Math Trainer Online Assessment and Intervention YOUR TURN Solve each sstem of equations b adding. Check our answers. 3. { + = -1 + = = - = 7. { 3 - = 1 5. { = 0 Unit 3

23 Solving a Linear Sstem b Subtracting If both equations contain the same - or -term, ou can solve b subtracting. EXAMPLE Solve the sstem of equations b subtracting. Check our answer = 6 { 3 - = -6 Math On the Spot STEP 1 Subtract the equations = 6 Write the equations so that like terms are aligned. -(3 - = -6) Notice that both equations contain the term 3. M Notes 0 + = 1 = 1 = 3 Subtract to eliminate the variable. Simplif and solve for. Divide each side b and simplif. STEP Substitute the solution into one of the original equations and solve for. 3 - = = -6 3 = -3 Use the second equation. Substitute 3 for the variable. Add 3 to each side. STEP 3 STEP = -1 Write the solution as an ordered pair: (-1, 3) Check the solution b graphing = = -6 -intercept: -intercept: - -intercept: -intercept: 6 The point of intersection is (-1, 3). Divide each side b 3 and simplif. - - O Reflect 6. What would happen if ou added the original equations? - - Lesson 8.3 5

24 7. How can ou decide whether to add or subtract to eliminate a variable in a linear sstem? Eplain our reasoning. YOUR TURN Personal Math Trainer Online Assessment and Intervention Solve each sstem of equations b subtracting. Check our answers. 6-3 = 6 8. { = = = { = { - 10 = 9 Math On the Spot Solving Problems with Sstems of Equations Man real-world situations can be modeled and solved with a sstem of equations. EXAMPLE 3 The Polar Bear Club wants to bu snowshoes and camp stoves. The club will spend $55.50 to bu them at Top Sports and $60.00 to bu them at Outdoor Eplorer, before taes, but Top Sports is farther awa. How man of each item does the club intend to bu? Image Credits: Jovan Nikolic/ Shutterstock Snowshoes Camp Stoves Top Sports $79.50 per pair $39.5 Outdoor Eplorer $89.00 per pair $ Unit 3

25 STEP 1 Choose variables and write a sstem of equations. Let represent the number of pairs of snowshoes. Let represent the number of camp stoves. Top Sports cost: = Outdoor Eplorer cost: = M Notes STEP Subtract the equations = ( = 60.00) = = = = 5 Both equations contain the term Subtract to eliminate the variable. Simplif and solve for. Divide each side b Simplif. STEP 3 Substitute the solution into one of the original equations and solve for = (5) = = = = = Use the first equation. Substitute 5 for the variable. Multipl. Subtract from each side. Divide each side b Simplif. STEP Write the solution as an ordered pair: (5, ) YOUR TURN The club intends to bu 5 pairs of snowshoes and camp stoves. 11. At the count fair, the Bater famil bought 6 hot dogs and juice drinks for $ The Farle famil bought 3 hot dogs and juice drinks for $ Find the price of a hot dog and the price of a juice drink. Personal Math Trainer Online Assessment and Intervention Lesson 8.3 7

26 Guided Practice 1. Solve the sstem { STEP = 1-3 = -11 b adding. (Eample 1) Add the equations. + 3 = = -11 Write the equations so that like terms are aligned. 5 + = 5 = Add to eliminate the variable. Simplif and solve for. = Divide both sides b and simplif. STEP Substitute into one of the original equations and solve for. = So, is the solution of the sstem. Solve each sstem of equations b adding or subtracting. (Eamples 1, ). { + = = = 3 3. { 3 - = 8. { - -5 = = -1 - = = = { 5 + = 1 6. { - + = 8 7. { = The Green River Freewa has a minimum and a maimum speed limit. Ton drove for hours at the minimum speed limit and 3.5 hours at the maimum limit, a distance of 355 miles. Rae drove hours at the minimum speed limit and 3 hours at the maimum limit, a distance of 30 miles. What are the two speed limits? (Eample 3)? a. Write equations to represent Ton s distance and Rae s distance. Ton: Rae: b. Solve the sstem. minimum speed limit: maimum speed limit: ESSENTIAL QUESTION CHECK-IN 9. Can ou use addition or subtraction to solve an sstem? Eplain. 8 Unit 3

27 Name Class Date 8.3 Independent Practice 10. Represent Real-World Problems Marta bought new fish for her home aquarium. She bought 3 guppies and platies for a total of $ Hank also bought guppies and platies for his aquarium. He bought 3 guppies and platies for a total of $ Find the price of a gupp and the price of a plat. Personal Math Trainer Online Assessment and Intervention 11. Represent Real-World Problems The rule for the number of fish in a home aquarium is 1 gallon of water for each inch of fish length. Marta s aquarium holds 13 gallons and Hank s aquarium holds 17 gallons. Based on the number of fish the bought in Eercise 10, how long is a gupp and how long is a plat? 1. Line m passes through the points (6, 1) and (, -3). Line n passes through the points (, 3) and (5, -6). Find the point of intersection of these lines. Image Credits: (tr) Johannes Kornelius/Shutterstock, (br) Corbis Flirt/Alam Images 13. Represent Real-World Problems Two cars got an oil change at the same auto shop. The shop charges customers for each quart of oil plus a flat fee for labor. The oil change for one car required 5 quarts of oil and cost $.5. The oil change for the other car required 7 quarts of oil and cost $5.5. How much is the labor fee and how much is each quart of oil? 1. Represent Real-World Problems A sales manager noticed that the number of units sold for two T-shirt stles, stle A and stle B, was the same during June and Jul. In June, total sales were $779 for the two stles, with A selling for $15.95 per shirt and B selling for $.95 per shirt. In Jul, total sales for the two stles were $385.10, with A selling at the same price and B selling at a discount of % off the June price. How man T-shirts of each stle were sold in June and Jul combined? 15. Represent Real-World Problems Adult tickets to a basketball game cost $5. Student tickets cost $1. A total of $,87 was collected on the sale of 1,6 tickets. How man of each tpe of ticket were sold? Lesson 8.3 9

28 FOCUS ON HIGHER ORDER THINKING Work Area 16. Communicate Mathematical Ideas Is it possible to solve the sstem { 3 - = 10 b using substitution? If so, eplain how. Which method, + = 6 substitution or elimination, is more efficient? Wh? 17. Jenn used substitution to solve the sstem { + = 8. Her solution is - = 1 shown below. Step 1 = Solve the first equation for. Step + (- + 8) = 8 Substitute the value of in an original equation. Step = 8 Use the Distributive Propert. Step 8 = 8 Simplif. a. Eplain the Error Eplain the error Jenn made. Describe how to correct it. b. Communicate Mathematical Ideas Would adding the equations have been a better method for solving the sstem? If so, eplain wh. 50 Unit 3

29 ? L E S S O N 8. Solving Sstems b Elimination with Multiplication ESSENTIAL QUESTION Students will solve a sstem of linear equations b multipling. How do ou solve a sstem of linear equations b multipling? Solving a Sstem b Multipling and Adding In some linear sstems, neither variable can be eliminated b adding or subtracting the equations directl. In sstems like these, ou need to multipl one of the equations b a constant so that adding or subtracting the equations will eliminate one variable. The steps for this method are as follows: Math On the Spot 1. Decide which variable to eliminate.. Multipl one equation b a constant so that adding or subtracting will eliminate that variable. 3. Solve the sstem using the elimination method. EXAMPLE 1 Solve the sstem of equations b multipling and adding = { 3-5 = -17 M Notes STEP 1 The coefficient of in the first equation, 10, is times the coefficient of, 5, in the second equation. Also, the -term in the first equation is being added, while the -term in the second equation is being subtracted. To eliminate the -terms, multipl the second equation b and add this new equation to the first equation. Multipl each term in the second equation b (3-5 = -17) to get opposite coefficients for the -terms = -3 Simplif = = Add the first equation to the new equation = -3 Add to eliminate the variable. 8 = -3 Simplif and solve for. 8 8 = -3 8 = - Divide each side b 8. Simplif. Lesson 8. 51

30 STEP Substitute the solution into one of the original equations and solve for = Use the first equation. (-) + 10 = = 10 = 10 = 1 Substitute - for the variable. Simplif. Add 8 to each side. Divide each side b 10 and simplif. Math Talk Mathematical Processes When ou check our answer algebraicall, wh do ou substitute our values for and into the original sstem? Eplain. STEP 3 STEP Write the solution as an ordered pair: (-, 1) Check our answer algebraicall. Substitute - for and 1 for in the original sstem = (-) + 10(1) = = { 3-5 = -17 3(-) - 5(1) = -1-5 = -17 The solution is correct. Reflect 1. How can ou solve this linear sstem b subtracting? Which is more efficient, adding or subtracting? Eplain our reasoning.. Can this linear sstem be solved b adding or subtracting without multipling? Wh or wh not? Personal Math Trainer Online Assessment and Intervention 3. What would ou need to multipl the second equation b to eliminate b adding? Wh might ou choose to eliminate instead of? YOUR TURN Solve each sstem of equations b multipling and adding. 5 + = = = -1. { = { 3 - = { + = 0 5 Unit 3

31 Solving a Sstem b Multipling and Subtracting You can solve some sstems of equations b multipling one equation b a constant and then subtracting. EXAMPLE Solve the sstem of equations b multipling and subtracting = 7 { - = -6 Math On the Spot M Notes STEP 1 Multipl the second equation b 3 and subtract this new equation from the first equation. Multipl each term in the second 3( - ) = -6 equation b 3 to get the same coefficients for the -terms. 6-1 = = 7 -(6-1 = -78) = = = = 5 Simplif. Subtract the new equation from the first equation. Subtract to eliminate the variable. Simplif and solve for. Divide each side b 17. Simplif. STEP STEP 3 STEP Substitute the solution into one of the original equations and solve for = (5) = = 7 6 = -18 = -3 Use the first equation. Substitute 5 for the variable. Simplif. Subtract 5 from each side. Divide each side b 6 and simplif. Write the solution as an ordered pair: (-3, 5) Check our answer algebraicall. Substitute -3 for and 5 for in the original sstem = 7 6(-3) + 5(5) = = 7 { - = -6 (-3) - (5) = -6-0 = -6 The solution is correct. Lesson 8. 53

32 Personal Math Trainer Online Assessment and Intervention YOUR TURN Solve each sstem of equations b multipling and subtracting = { 3-7 = 6-9 = 9 8. { = -11 { 9 + = = -11 Math On the Spot M Notes Solving Problems with Sstems of Equations Man real-world situations can be modeled with a sstem of equations. EXAMPLE 3 Problem Solving The Simon famil attended a concert and visited an art museum. Concert tickets were $.75 for adults and $16.00 for children, for a total cost of $ Museum tickets were $8.5 for adults and $.50 for children, for a total cost of $.75. How man adults and how man children are in the Simon famil? Analze Information The answer is the number of adults and children. Formulate a Plan Solve a sstem to find the number of adults and children. Justif Solve and Evaluate STEP 1 STEP Choose variables and write a sstem of equations. Let represent the number of adults. Let represent the number of children. Concert cost: = Museum cost: =.75 Multipl both equations b 100 to eliminate the decimals. 100( = 138.5),75 + 1,600 = 13,85 Image Credits: Education Images/UIG/Gett Images 100( =.75) =,75 5 Unit 3

33 STEP 3 Multipl the second equation b 3 and subtract this new equation from the first equation. 3( =,75),75 + 1,350 = 1,85 Multipl each term in the second equation b 3 to get the same coefficients for the -terms. Simplif.,75 + 1,600 = 13,85 -(,75 + 1,350 = 1,85) = 1, = 1, = 1, = Subtract the new equation from the first equation. Subtract to eliminate the variable. Simplif and solve for. Divide each side b 50. Simplif. STEP Substitute the solution into one of the original equations and solve for = () = = =.75 = 3 Use the second equation. Substitute for the variable. Simplif. Subtract 18 from each side. Divide each side b 8.5 and simplif. STEP 5 Write the solution as an ordered pair: (3, ). There are 3 adults and children in the famil. Justif and Evaluate Substituting = 3 and = into the original equations results in true statements. The answer is correct. YOUR TURN 10. Contestants in the Run-and-Bike-a-thon run for a specified length of time, then bike for a specified length of time. Jason ran at an average speed of 5. mi/h and biked at an average speed of 0.6 mi/h, going a total of 1. miles. Seth ran at an average speed of 10. mi/h and biked at an average speed of 18. mi/h, going a total of 17 miles. For how long do contestants run and for how long do the bike? Personal Math Trainer Online Assessment and Intervention Lesson 8. 55

34 Guided Practice 3 - = 8 1. Solve the sstem { - + = -1 b multipling and adding. (Eample 1) STEP 1 Multipl the first equation b. Add to the second equation. Multipl each term in the first equation b to get (3 - = 8) opposite coefficients for the -terms. - = + (-) + = = Simplif. Add the second equation to the new equation. Add to eliminate the variable. = Divide both sides b and simplif. STEP Substitute into one of the original equations and solve for. = So, is the solution of the sstem. Solve each sstem of equations b multipling first. (Eamples 1, ). { + = 3 + = = = 7 3. {. + 3 = 18 { 6 - = 1 5. { + = = = = { + 3 = 5 7. { = 0 8. Brce spent $5.6 on some apples priced at $0.6 each and some pears priced at $0.5 each. At another store he could have bought the same number of apples at $0.3 each and the same number of pears at $0.39 each, for a total cost of $3.6. How man apples and how man pears did Brce bu? (Eample 3)? a. Write equations to represent Brce s ependitures at each store. Second store: First store: b. Solve the sstem. Number of apples: ESSENTIAL QUESTION CHECK-IN Number of pears: 9. When solving a sstem b multipling and then adding or subtracting, how do ou decide whether to add or subtract? 56 Unit 3

35 Name Class Date 8. Independent Practice Personal Math Trainer Online Assessment and Intervention 10. Eplain the Error Gwen used elimination with multiplication to solve the sstem + 6 = 3 { - 3 = -1. Her work to find is shown. Eplain her error. Then solve the sstem. ( - 3) = -1-6 = = = = 1_ 11. Represent Real-World Problems At Raging River Sports, polester-fill sleeping bags sell for $79. Down-fill sleeping bags sell for $19. In one week the store sold 1 sleeping bags for $1,56. Sleeping Bags a. Let represent the number of polester-fill bags sold and let represent the number of down-fill bags sold. Write a sstem of equations ou can solve to find the number of each tpe sold. Nlon $19 Flannel-lined $79 b. Eplain how ou can solve the sstem for b multipling and subtracting. c. Eplain how ou can solve the sstem for using substitution. d. How man of each tpe of bag were sold? 1. Twice a number plus twice a second number is 310. The difference between the numbers is 55. Find the numbers b writing and solving a sstem of equations. Eplain how ou solved the sstem. Lesson 8. 57

36 13. Represent Real-World Problems A farm stand sells apple pies and jars of applesauce. The table shows the number of apples needed to make a pie and a jar of applesauce. Yesterda, the farm picked 169 Grann Smith apples and 95 Red Delicious apples. How man pies and jars of applesauce can the farm make if ever apple is used? Tpe of apple Grann Smith Red Delicious Needed for a pie 5 3 Needed for a jar of applesauce FOCUS ON HIGHER ORDER THINKING Work Area 1. Make a Conjecture Lena tried to solve a sstem of linear equations algebraicall and in the process found the equation 5 = 9. Lena thought something was wrong, so she graphed the equations and found that the were parallel lines. Eplain what Lena s graph and equation could mean. + 3 = Consider the sstem { = -1. a. Communicate Mathematical Ideas Describe how to solve the sstem b multipling the first equation b a constant and subtracting. Wh would this method be less than ideal? b. Draw Conclusions Is it possible to solve the sstem b multipling both equations b integer constants? If so, eplain how. Image cedits: Photodisc/Gett Images c. Use our answer from part b to solve the sstem. 58 Unit 3

37 L E S S O N 8.5 Solving Special Sstems Students will solve sstems with no solution or infinitel man solutions.? ESSENTIAL QUESTION How do ou solve sstems with no solution or infinitel man solutions? EXPLORE ACTIVITY Solving Special Sstems b Graphing As with equations, some sstems ma have no solution or infinitel man solutions. One wa to tell how man solutions a sstem has is b inspecting its graph. Use the graph to solve each sstem of linear equations. A { + = 7 + = 6 Is there a point of intersection? Eplain = 3 + = O = 1 + = 7 6 Does this linear sstem have a solution? Use the graph to eplain. B { + = 6 + = 3 Is there a point of intersection? Eplain. Does this linear sstem have a solution? Use the graph to eplain. Reflect 1. Use the graph to identif two lines that represent a linear sstem with eactl one solution. What are the equations of the lines? Eplain our reasoning. Lesson

38 EXPLORE ACTIVITY (cont d). If each equation in a sstem of two linear equations is represented b a different line when graphed, what is the greatest number of solutions the sstem can have? Eplain our reasoning. 3. Identif the three possible numbers of solutions for a sstem of linear equations. Eplain when each tpe of solution occurs. Math On the Spot Solving Special Sstems Algebraicall As with equations, if ou solve a sstem of equations with no solution, ou get a false statement, and if ou solve a sstem with infinitel man solutions, ou get a true statement. EXAMPLE 1 M Notes A Solve the sstem of linear equations b substitution. { - = = STEP 1 STEP STEP 3 STEP Solve - = - for : = - Substitute the resulting epression into the other equation and solve. Substitute the -( - ) + = epression for 8 the variable. = Simplif. Interpret the solution. The result is the false statement =, which means there is no solution. Graph the equations to check our answer. The graphs do not intersect, so there is no solution O - 60 Unit 3

39 B Solve the sstem of linear equations b elimination. + = - { + = - STEP 1 STEP Multipl the first equation b -. -( + = -) - + (-) = Add the new equation from Step 1 to the original second equation. - + (-) = + + = = O 0 = 0 - Math Talk Mathematical Processes What solution do ou get when ou solve the sstem in part B b substitution? Does this result change the number of solutions? Eplain. STEP 3 Interpret the solution. The result is the statement 0 = 0, which is alwas true. This means that the sstem has infinitel man solutions STEP Graph the equations to check our answer. The graphs are the same line, so there are infinitel man solutions. Reflect. If represents a variable and a and b represent constants so that a b, interpret what each result means when solving a sstem of equations. = a a = b a = a 5. In part B, can ou tell without solving that the sstem has infinitel man solutions? If so, how? YOUR TURN Solve each sstem. Tell how man solutions each sstem has. - 6 = 9 6. { = 7. { + = = = 6 8. { -3 + = 1 Personal Math Trainer Online Assessment and Intervention Lesson

40 Guided Practice 1. Use the graph to solve each sstem of linear equations. (Eplore Activit) - = -6 - = -6 A. { - = B. { + = 6 C. { - = 6-3 = 1 STEP 1 Decide if the graphs of the equations in each sstem intersect, are parallel, or are the same line. 6 STEP Sstem A: The graphs. Sstem B: The graphs. Sstem C: The graphs. Decide how man points the graphs have in common. Intersecting lines have point(s) in common. - = = 6-3 = 1 + = 6 O 6 STEP 3 Parallel lines have The same lines have Solve each sstem. point(s) in common. point(s) in common. Sstem A has points in common, so it has solution. Sstem B has point in common. That point is the solution,. Sstem C has points in common. ordered pairs on the line will make both equations true. Solve each sstem. Tell how man solutions each sstem has. (Eample 1) - 3 = 6 + = = -10. { = { 3 + =. { 3 + = -5? ESSENTIAL QUESTION CHECK-IN 5. When ou solve a sstem of equations algebraicall, how can ou tell whether the sstem has zero, one, or an infinite number of solutions? 6 Unit 3

41 Name Class Date 8.5 Independent Practice Personal Math Trainer Online Assessment and Intervention Solve each sstem b graphing. Check our answer algebraicall = = 5 6. { - 3 = 3 7. { 3 + = O - - O Solution: Solution: For Es. 8 1, state the number of solutions for each sstem of linear equations. 8. a sstem whose graphs have the same slope but different -intercepts 9. a sstem whose graphs have the same -intercepts but different slopes 10. a sstem whose graphs have the same -intercepts and the same slopes 11. a sstem whose graphs have different -intercepts and different slopes 1. the sstem { = = the sstem { = = the sstem whose graphs were drawn using these tables of values: Equation 1 Equation Draw Conclusions The graph of a linear sstem appears in a tetbook. You can see that the lines do not intersect on the graph, but also the do not appear to be parallel. Can ou conclude that the sstem has no solution? Eplain. Lesson

42 16. Represent Real-World Problems Two school groups go to a roller skating rink. One group pas $3 for 36 admissions and 1 skate rentals. The other group pas $81 for 1 admissions and 7 skate rentals. Let represent the cost of admission and let represent the cost of a skate rental. Is there enough information to find values for and? Eplain. 17. Represent Real-World Problems Juan and Tor are practicing for a track meet. The start their practice runs at the same point, but Tor starts 1 minute after Juan. Both run at a speed of 70 feet per minute. Does Tor catch up to Juan? Eplain. FOCUS ON HIGHER ORDER THINKING Work Area 18. Justif Reasoning A linear sstem with no solution consists of the equation = - 3 and a second equation of the form = m + b. What can ou sa about the values of m and b? Eplain our reasoning. 19. Justif Reasoning A linear sstem with infinitel man solutions consists of the equation = 8 and a second equation of the form A + B = C. What can ou sa about the values of A, B, and C? Eplain our reasoning. 0. Draw Conclusions Both the points (, -) and (, -) are solutions of a sstem of linear equations. What conclusions can ou make about the equations and their graphs? Image Credits: Bill Brooks/ Alam Images 6 Unit 3

43 MODULE QUIZ Read 8.1 Solving Sstems of Linear Equations b Graphing Solve each sstem b graphing. Personal Math Trainer Online Assessment and Intervention 1. { = - 1 = - 3. { + = = - - O - - O Solving Sstems b Substitution Solve each sstem of equations b substitution. 3. { = + = Solving Sstems b Elimination 3 - = 11. { + = 9 Solve each sstem of equations b adding or subtracting. 5. { 3 + = = + = 5 6. { 3 + = 8. Solving Sstems b Elimination with Multiplication Solve each sstem of equations b multipling first. + 3 = = 7. { 3 + = { 3 - = Solving Special Sstems Solve each sstem. Tell how man solutions each sstem has = = { - = { + 3 = - ESSENTIAL QUESTION 11. What are the possible solutions to a sstem of linear equations, and what do the represent graphicall? Module 8 65

44 MODULE 8 MIXED REVIEW Assessment Readiness Personal Math Trainer Online Assessment and Intervention Selected Response 1. The graph of which equation is shown? O A = - + C = + B = - + D = + 1. Which best describes the solutions to the sstem { + = = 0? A one solution B no solution D (0, 0) C infinitel man 3. Which of the following represents written in scientific notation? A C B D Which is the solution to { - = 1 + = 11? A (, 3) C (-, 3) B (3, ) D (3, -) 5. Which epression can ou substitute in the indicated equation to solve { 3 - = 5 + =? A - for in 3 - = 5 B - for in 3 - = 5 C 3-5 for in 3 - = 5 D 3-5 for in + = 6. What is the solution to the sstem of linear equations shown on the graph? O A -1 C (-1, -) B - D (-, -1) 7. Which step could ou use to start solving { - 6 = 8-5 = 3? Mini-Task A Add - 5 = 3 to - 6 = 8. B Multipl - 6 = 8 b and add it to - 5 = 3. C Multipl - 6 = 8 b and subtract it from - 5 = 3. D Substitute = 6-8 for in - 5 = A hot-air balloon begins rising from the ground at meters per second at the same time a parachutist s chute opens at a height of 00 meters. The parachutist descends at 6 meters per second. a. Define the variables and write a sstem that represents the situation. b. Find the solution. What does it mean? 66 Unit 3