Solving Special Systems of Linear Equations


 Ellen Lloyd
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1 Solving Sstems of Linear Equations Solving Sstems of Linear Equations b Graphing Solving Sstems of Linear Equations b Substitution Solving Sstems of Linear Equations b Elimination Solving Special Sstems of Linear Equations Sstems of Linear Inequalities ar em of line raph a sst e number of g u o n a C s th that show oing equations d treats that I am g n a s it u? c u bis ith o to share w He look over here. Ca n ou estimate the solution of the sstem of linear equations that I made with these ca ttails? Copright Big Ideas Learning, LLC MSCC_ALG_PE_000_co.indd 5 /9/ :: PM
2 What You Learned Before Hold ou r tail a bit lower. Eample Solve ( + ) = 6. ( + ) = 6 6 = 6 Write the equation. Use Distributive Propert. = 6 Combine like terms. = 8 Add to each side. = 9 Divide each side b. The solution is = 9. Solve the equation. Check our solution = 7. 7w + w 5 = 7. (z 8) + 0 = 5. = 0c (c 9) Eample Step : Graph + in a coordinate plane. Graph + =. Use a solid line because the inequalit smbol is. Step : Test (0, 0). +? (0) Step : Write the inequalit. Substitute. Ź Ź Ź (0, 0) Ź Ź Simplif. Because (0, 0) is a solution, shade the halfplane that contains (0, 0). Graph the inequalit in a coordinate plane. 5. < > Copright Big Ideas Learning, LLC MSCC_ALG_PE_000_co.indd 5 /9/ :: PM
3 . Solving Sstems of Linear Equations b Graphing How can ou solve a sstem of linear equations? ACTIVITY: Writing a Sstem of Linear Equations Work with a partner. Your famil starts a bedandbreakfast. The spend $500 fiing up a bedroom to rent. The cost for food and utilities is $0 per night. Your famil charges $60 per night to rent the bedroom. a. Write an equation that represents the costs. Cost, C (in dollars) = $0 per night Number of nights, + $500 b. Write an equation that represents the revenue (income). Revenue, R (in dollars) = $60 per night Number of nights, c. A set of two (or more) linear equations is called a sstem of linear equations. Write the sstem of linear equations for this problem. ACTIVITY: Using a Table to Solve a Sstem COMMON CORE Sstems of Equations In this lesson, ou will write and solve sstems of linear equations b graphing. solve reallife problems. Learning Standards 8.EE.8a 8.EE.8b 8.EE.8c A.CED. A.REI.6 Use the cost and revenue equations from Activit to find how man nights our famil needs to rent the bedroom before recovering the cost of fiing up the bedroom. This is the breakeven point. a. Cop and complete the table C R b. How man nights does our famil need to rent the bedroom before breaking even? 5 Chapter Solving Sstems of Linear Equations Copright Big Ideas Learning, LLC
4 ACTIVITY: Using a Graph to Solve a Sstem a. Graph the cost equation from Activit. 600 b. In the same coordinate plane, graph the revenue equation from Activit. c. Find the point of intersection of the two graphs. What does this point represent? How does this compare to the breakeven point in Activit? Eplain EXAMPLE: Using a Graphing Calculator Math Practice Use Technolog to Eplore How do ou decide the values for the viewing window of our calculator? What other viewing windows could ou use? Use a graphing calculator to solve the sstem. = Equation = 60 Equation a. Enter the equations into our calculator. Then graph the equations in an appropriate window. b. To find the solution, use the intersect feature to find the point of intersection. The solution is (0, 600) IN YOUR OWN WORDS How can ou solve a sstem of linear equations? How can ou check our solution? 6. Solve one of the sstems b using a table, another sstem b sketching a graph, and the remaining sstem b using a graphing calculator. Eplain wh ou chose each method. a. =. +. b. = c. = 5 =.7. = + 9 = + Use what ou learned about sstems of linear equations to complete Eercises 6 on page 58. Copright Big Ideas Learning, LLC Section. Solving Sstems of Linear Equations b Graphing 55
5 Lesson. Lesson Tutorials A sstem of linear equations is a set of two or more linear equations in the same variables. An eample is shown below. Ke Vocabular sstem of linear equations, p. 56 solution of a sstem of linear equations, p. 56 =+ Equation = 7 Equation A solution of a sstem of linear equations in two variables is an ordered pair that is a solution of each equation in the sstem. The solution of a sstem of linear equations is the point of intersection of the graphs of the equations. Reading Solving a Sstem of Linear Equations b Graphing Step Graph each equation in the same coordinate plane. A sstem of linear equations is also called a linear sstem. Step Estimate the point of intersection. Step Check the point from Step b substituting for and in each equation of the original sstem. EXAMPLE Solving a Sstem of Linear Equations b Graphing Solve the sstem b graphing. = + 5 Equation = Equation Step : Graph each equation. Step : Estimate the point of intersection. The graphs appear to intersect at (, ). 6 Step : Check the point from Step. Check 6 â Ź Ź â à 5 Equation Equation = + 5? = ( ) + 5 =? = ( ) = Ź5 = â Ź Ź 5 â à 5 (Ź, ) Ź5 Ź Ź Ź The solution is (, ). Ź Solve the sstem of linear equations b graphing. Eercises 0. =. = + 56 Chapter MSCC_ALG_PE_00.indd 56 Solving Sstems of Linear Equations = 5 + = 0. = = + Copright Big Ideas Learning, LLC /9/ :6:50 PM
6 EXAMPLE RealLife Application A kicker on a football team scores point for making an etra point and points for making a field goal. The kicker makes a total of 8 etra poi nts and field goals in a game and scores points. Write and solve a ss tem of linear equations to find the number of etra points and the number of field goals. Use a verbal model to write a sstem of linear equations. Number of etra points, + Number of field goals, = Total number of kicks Points per etra point Number of etra + points, Points per field goal Number of field goals, = Total number of points Stud Tip It ma be easier to graph the equations in a sstem b rewriting the equations in slopeintercept form. The sstem is: + = 8 Equation + = Equation Step : Graph each equation. Step : Estimate the point of intersection. The graphs appear to intersect at (6, ). Step : Check our point from Step. Equation Equation + = 8 + = 6 + =? () =? 8 = 8 = The solution is (6, ). So, the kicker made 6 etra points and field goals Check 8 8 (6, ) Eercises 5 Solve the sstem of linear equations b graphing.. = 7 5. = = 6 + = + = 6 + = 8 7. WHAT IF? In Eample, the kicker makes a total of 7 etra points and field goals and scores 7 points. Write and solve a sstem of linear equations to find the numbers of etra points and field goals. Copright Big Ideas Learning, LLC Section. Solving Sstems of Linear Equations b Graphing 57
7 . Eercises Help with Homework. VOCABULARY Do the equations = 5 and 7 + = 8 form a sstem of linear equations? Eplain.. WRITING What does it mean to solve a sstem of equations?. WRITING You graph a sstem of linear equations and the solution appears to be (, ). How can ou verif that the solution is (, )? 9+(6)= +()= +(9)= 9+()= Use a table to find the breakeven point. Check our solution.. C = C = C = R = 5 R = R = 76 Match the sstem of linear equations with the corresponding graph. Use the graph to estimate the solution. Check our solution. 7. =.5 8. = + 9. = = + = = + 5 A B C Solve the sstem of linear equations b graphing. 0. = + 9. =. = + 5 = 6 = 5 + =. + = 7. = 7 5. = 7 = + = = 5 6. CARRIAGE RIDES The cost C (in dollars) for the care and maintenance of a horse and carriage is C = , where is the number of rides. a. Write an equation for the revenue R in terms of the number of rides. b. How man rides are needed to break even? 58 Chapter Solving Sstems of Linear Equations Copright Big Ideas Learning, LLC
8 Use a graphing calculator to solve the sstem of linear equations = = =.. = = =. 0. ERROR ANALYSIS Describe and correct the error in solving the sstem of linear equations.. REASONING Is it possible for a sstem of two linear equations to have eactl two solutions? Eplain our reasoning The solution of the linear sstem = and = + 7 is =.. MODELING You have a total of math and science problems for homework. You have 0 more math problems than science problems. How man problems do ou have in each subject? Use a sstem of linear equations to justif our answer.. CANOE RACE You and our friend are in a canoe race. Your friend is a half mile in front of ou and paddling miles per hour. You are paddling. miles per hour. our friend ou a. You are 8.5 miles from the finish line. How long will it take ou to catch up to our friend? b. You both maintain our paddling rates for the remainder of the race. How far ahead of our friend will ou be when ou cross the finish line?. Your friend is tring to grow her hair as long as her cousin s hair. The table shows their hair lengths (in inches) in different months. Month Friend s Hair (in.) Cousin s Hair (in.) a. Write a sstem of linear equations that represents this situation. b. Will our friend s hair ever be as long as her cousin s hair? If so, in what month? Solve the equation. Check our solution. (Section.) 5. c c + = ( ) + = ( + 8) = 9 8. MULTIPLE CHOICE The graph of which equation is perpendicular to the graph of = +? (Section.) A = B = + C = D = + Copright Big Ideas Learning, LLC Section. Solving Sstems of Linear Equations b Graphing 59
9 . Solving Sstems of Linear Equations b Substitution How can ou use substitution to solve a sstem of linear equations? ACTIVITY: Using Substitution to Solve a Sstem Work with a partner. Solve each sstem of linear equations using two methods. Method : Solve for first. Solve for in one of the equations. Use the epression for to find the solution of the sstem. Eplain how ou did it. Method : Solve for first. Solve for in one of the equations. Use the epression for to find the solution of the sstem. Eplain how ou did it. Is the solution the same using both methods? a. 6 = b. = c. + = 5 + = 7 = 5 = d. 5 = e. + = f. 6 = 6 6 = 5 + = 7 8 = 5 ACTIVITY: Writing and Solving a Sstem of Equations COMMON CORE Sstems of Equations In this lesson, ou will write and solve sstems of linear equations b substitution. solve reallife problems. Learning Standards 8.EE.8b 8.EE.8c A.CED. A.REI.6 Work with a partner. a. Roll a pair of number cubes that have different colors. Then write the ordered pair shown b the number cubes. The ordered pair at the right is (, ). b. Write a sstem of linear equations that has this ordered pair as its solution. value c. Echange sstems with our partner and use one of the methods from Activit to solve the sstem. value 60 Chapter Solving Sstems of Linear Equations Copright Big Ideas Learning, LLC
10 ACTIVITY: Solving a Secret Code Math Practice Check Progress As ou complete each sstem of equations, how do ou know our answer is correct? Work with a partner. Decode the quote b Archimedes , (A, C) + = (D, E) + = 0 (G, H) + = 0 = = 0 = 6 (I, L) + = 9 (M, N) + = (O, P) + = = = = 6 (R, S) + = (T, U) + = 7 (V, W) + = 0 = 6 = 0 =. IN YOUR OWN WORDS How can ou use substitution to solve a sstem of linear equations? Use what ou learned about sstems of linear equations to complete Eercises 6 on page 6. Copright Big Ideas Learning, LLC Section. Solving Sstems of Linear Equations b Substitution 6
11 . Lesson Lesson Tutorials Another wa to solve sstems of linear equations is to use substitution. Solving a Sstem of Linear Equations b Substitution Step Solve one of the equations for one of the variables. Step Substitute the epression from Step into the other equation and solve for the other variable. Step Substitute the value from Step into one of the original equations and solve. Check Equation = EXAMPLE 6 =? ( ) 6 = 6 Equation 7 = 5 7( ) ( 6) =? 5 5 = 5 Solving a Sstem of Linear Equations b Substitution Solve the sstem b substitution. = Equation 7 = 5 Equation Step : Equation is alread solved for. Step : Substitute for in Equation. 7 = 5 Equation 7 ( ) = 5 Substitute for = 5 Use the Distributive Propert. + 8 = 5 Combine like terms. = Subtract 8 from each side. = Divide each side b. Step : Substitute for in Equation and solve for. = Equation = ( ) Substitute for. = Multipl. = 6 Subtract. The solution is (, 6). Eercises 0 5 Solve the sstem of linear equations b substitution. Check our solution.. = +. + = 0. = 5 + = 5 = 5 + = 6 Chapter Solving Sstems of Linear Equations Copright Big Ideas Learning, LLC
12 EXAMPLE RealLife Application You bu a total of 50 turke burgers and veggie burgers for $90. You pa $ per turke burger and $.50 per veggie burger. Write and solve a sstem of linear equations to find the number of turke burgers and the number of veggie burgers ou bu. Use a verbal model to write a sstem of linear equations. Number of turke burgers, + Number of veggie burgers, = Total number of burgers Cost per turke burger Number of turke burgers, + Cost per veggie burger Number of veggie burgers, = Total cost The sstem is: + = 50 Equation +.5 = 90 Equation Step : Solve Equation for. Stud Tip It is easiest to solve for a variable that has a coefficient of or. + = 50 Equation = 50 Subtract from each side. Step : Substitute 50 for in Equation. +.5 = 90 Equation (50 ) +.5 = 90 Substitute 50 for. Check = 90 Use the Distributive Propert. 0.5 = 0 Simplif = 0 Divide each side b 0.5. Step : Substitute 0 for in Equation and solve for = 50 Equation = 50 Substitute 0 for. = 0 Subtract 0 from each side. You bu 0 turke burgers and 0 veggie burgers. Eercises 8 0. A juice stand sells lemonade for $ per cup and orange juice for $ per cup. The juice stand sells a total of 00 cups of juice for $0. Write and solve a sstem of linear equations to find the number of cups of lemonade and the number of cups of orange juice sold. Copright Big Ideas Learning, LLC Section. Solving Sstems of Linear Equations b Substitution 6
13 Eercises. Help with Homework. WRITING Describe how to solve a sstem of linear equations b substitution.. NUMBER SENSE When solving a sstem of linear equations b substitution, how do ou decide which variable to solve for in Step?. REASONING Does solving a sstem of linear equations b graphing give the same solution as solving b substitution? Eplain our reasoning. 6)= 9+( )= +( 9)= +( = ) 9+( Write a sstem of linear equations that has the ordered pair as its solution. Use a method from Activit to solve the sstem Tell which equation ou would use in Step when solving the sstem b substitution. Eplain our reasoning = 5 = = = + 6 = = Solve the sstem of linear equations b substitution. Check our solution. 0. = = 0. = =. = + 5 =. = 0 +7=. = + 7 = 5. 8 = 0 + = 6. SCHOOL CLUBS There are a total of 6 students in a drama club and a earbook club. The drama club has 0 more students than the earbook club. esents this situation. a. Write a sstem of linear equations that represents b. How man students are in the drama club? the earbook club? 7. THEATER A drama club earns $00 from a production. A total of 6 adult tickets and student tickets are sold. An adult ticket costs twice as much as a student ticket. a. Write a sstem of linear equations that represents this situation. b. What is the cost of each ticket? 6 Chapter Solving Sstems of Linear Equations Copright Big Ideas Learning, LLC MSCC_ALG_PE_00.indd 6 /9/ :7:5 PM
14 Solve the sstem of linear equations b substitution. Check our solution. 8. = = 0. 5 = 5 = = + 8 = 6. ERROR ANALYSIS Describe and correct the error in solving the sstem of linear equations. + = 5 Equation Step : Step : = Equation + = 5 + ( + 5) = 5 = = 5 5 = 5. STRUCTURE The measure of the obtuse angle in the isosceles triangle is two and a half times the measure of one base angle. Write and solve a sstem of linear equations to find the measures of all the angles.. ANIMAL SHELTER An animal shelter has a total of 65 abandoned cats and dogs. The ratio of cats to dogs is 6 : 7. How man cats are in the shelter? How man dogs are in the shelter? Justif our answers.. NUMBER SENSE The sum of the digits of a twodigit number is 8. When the digits are reversed, the number increases b 6. Find the original number. 5. Repeated Reasoning A DJ has a total of 075 dance, rock, and countr songs on her sstem. The dance selection is three times the size of the rock selection. The countr selection has 05 more songs than the rock selection. How man songs on the sstem are dance? rock? countr? Write the equation in standard form. (Section.) 6. 9 = = 8. 6 = + 9. MULTIPLE CHOICE Use the figure to find the measure of. (Skills Review Handbook) A 7 B 7 C 8 D t a b Section. Solving Sstems of Linear Equations b Substitution 65
15 Stud Help Graphic Organizer You can use a notetaking organizer to write notes, vocabular, and questions about a topic. Here is an eample of a notetaking organizer for solving sstems of linear equations b graphing. Write important vocabular or formulas in this space. sstem of linear equations solution of a sstem of linear equations Solving sstems of linear equations b graphing Step : Graph each equation. Step : Estimate the point of intersection. Step : Check the point from Step. Write our notes about the topic in this space. Eample: Solve the sstem. The solution is (, 0). Will a sstem of linear equations alwas have a solution? = + = (, 0) = + = Write our questions about the topic in this space. Make a notetaking organizer to help ou stud this topic.. solving sstems of linear equations b substitution After ou complete this chapter, make notetaking organizers for the following topics.. solving sstems of linear equations b elimination. graphing sstems of linear inequalities M notetaking organizer has me thinking about retirement when I won t have to fetch sticks anmore. 66 Chapter Solving Sstems of Linear Equations Copright Big Ideas Learning, LLC
16 .. Quiz Progress Check Match the sstem of linear equations with the corresponding graph. Use the graph to estimate the solution. Check our solution. (Section.). =. =. = A. = + = + = B. C. Solve the sstem of linear equations b graphing. (Section.). = = 6. + = = + 9 = + = Solve the sstem of linear equations b substitution. Check our solution. (Section.) 7. = 8 8. = = = 5 = + 9 = 0. MOVIE CLUB Members of a movie rental club pa a $5 annual membership fee and $ for new release movies. Nonmembers pa $ for new release movies. (Section.) a. Write a sstem of linear equations that represents this situation. b. When is it beneficial to have a membership?. NUMBER SENSE The sum of two numbers is 8. The greater number is 8 more than the other number. Find each number. Use a sstem of linear equations to justif our answer. (Section.). VOLLEYBALL The length of a sand volleball court is twice its width. The perimeter is 80 feet. Find the length and width of the sand volleball court. (Section.). MEDICAL STAFF A hospital emplos a total of 77 nurses and doctors. The ratio of nurses to doctors is 9 :. How man nurses are emploed at the hospital? How man doctors are emploed at the hospital? (Section.) Copright Big Ideas Learning, LLC Sections.. Quiz 67
17 . Solving Sstems of Linear Equations b Elimination How can ou use elimination to solve a sstem of linear equations? ACTIVITY: Using Elimination to Solve a Sstem Work with a partner. Solve each sstem of linear equations using two methods. Method : Subtract. Subtract Equation from Equation. What is the result? Eplain how ou can use the result to solve the sstem of equations. Method : Add. Add the two equations. What is the result? Eplain how ou can use the result to solve the sstem of equations. Is the solution the same using both methods? a. + = b. = c. + = 7 = 0 + = = 5 ACTIVITY: Using Elimination to Solve a Sstem Work with a partner. + = Equation COMMON CORE Sstems of Equations In this lesson, ou will write and solve sstems of linear equations b elimination. solve reallife problems. Learning Standards 8.EE.8b 8.EE.8c A.CED. A.REI.5 A.REI = Equation a. Can ou add or subtract the equations to solve the sstem of linear equations? Eplain. b. Eplain what propert ou can appl to Equation in the sstem so that the coefficients are the same. c. Eplain what propert ou can appl to Equation in the sstem so that the coefficients are the same. d. You solve the sstem in part (b). Your partner solves the sstem in part (c). Compare our solutions. e. Use a graphing calculator to check our solution. 68 Chapter Solving Sstems of Linear Equations Copright Big Ideas Learning, LLC
18 ACTIVITY: Solving a Secret Code Math Practice Find Entr Points What is the first thing ou do to solve a sstem of linear equations b elimination? Wh? Work with a partner. Solve the puzzle to find the name of a famous mathematician who lived in Egpt around 50 A.D. B W R M F Y K N O J A S I D X Z Q P C E G B T J M R C Z N O U W 0 K X U H L Y S Q F E A S W K R M G J Z N H V D G E L X L F Q O B IN YOUR OWN WORDS How can ou use elimination to solve a sstem of linear equations? 5. When can ou add or subtract equations in a sstem to solve the sstem? When do ou have to multipl first? Justif our answers with eamples. 6. In Activit, wh can ou multipl equations in the sstem b a constant and not change the solution of the sstem? Eplain our reasoning. Use what ou learned about sstems of linear equations to complete Eercises 6 on page 7. Copright Big Ideas Learning, LLC Section. Solving Sstems of Linear Equations b Elimination 69
19 Lesson. Lesson Tutorials Solving a Sstem of Linear Equations b Elimination Step Multipl, if necessar, one or both equations b a constant so at least one pair of like terms has the same or opposite coefficients. Step Add or subtract the equations to eliminate one of the variables. Step Solve the resulting equation for the remaining variable. Step Substitute the value from Step into one of the original equations and solve. EXAMPLE Solving a Sstem of Linear Equations b Elimination Solve the sstem b elimination. Stud Tip Because the coefficients of are the same, ou can also solve the sstem b subtracting in Step. + = = 6 6 = 8 So, =. = Step : Add the equations. + = Equation = 6 Equation = Add the equations. = Equation from Step =7 Divide each side b. + = Equation 7 + = Substitute 7 for. = 9 Equation Subtract 7 from each side. = Eercises 7 Divide each side b. The solution is (7, ). Solve the sstem of linear equations b elimination. Check our solution.. = 9. + = Chapter MSCC_ALG_PE_00.indd 70 Equation Step : Substitute 7 for in one of the original equations and solve for. + =? 7 + ( ) = 70 = 6 Step : Solve for. Equation 6 = 6 Equation Step : The coefficients of the terms are alread opposites. Check = 6? 7 ( ) = 6 + = Solving Sstems of Linear Equations 5 + = 5 + =. + = = Copright Big Ideas Learning, LLC /9/ :8:6 PM
20 EXAMPLE Solving a Sstem of Linear Equations b Elimination Solve the sstem b elimination = 5 Equation = Equation Step : Multipl Equation b = = 5 Equation = Multipl b. 6 = Revised Equation Stud Tip In Eample, notice that ou can also multipl Equation b and then add the equations. Step : Subtract the equations = 5 Equation 6 = Revised Equation Step : Solve for. 7 = 7 Subtract the equations. 7 = 7 Equation from Step = Divide each side b 7. Step : Substitute for in one of the original equations and solve for. = Equation ( ) = Substitute for. + = = 0 Multipl. Subtract from each side. = 5 Divide each side b. The solution is ( 5, ). Check Solve the sstem of linear equations b elimination. Check our solution.. + = 5. 5 = = = = 8 = + Copright Big Ideas Learning, LLC Section. Solving Sstems of Linear Equations b Elimination 7
21 EXAMPLE RealLife Application You bu 8 hostas and 5 dalilies for $9. Your friend bus hostas and dalilies for $7. Write and solve a sstem of linear equations to find the cost of each dalil. Use a verbal model to write a sstem of linear equations. Nu Number h of hostas of each Cost hosta, + The sstem is: = 9 + = 7 Number of dalilies Cost of each = dalil, Total cost Equation (You) Equation (Your friend) Step : To find the cost of each dalil, eliminate the terms. Multipl Equation b. Multipl Equation b = 9 Multipl b. + 5 = 579 Revised Equation + = 7 Multipl b = 96 Revised Equation Step : Subtract the revised equations. + 5 = 579 Revised Equation + 96 = 96 Revised Equation 5 = 57 Subtract the equations. Step : Solving the equation 5 = 57 gives = 7. Each dalil costs $7. Eercises 6 7. A landscaper bus peonies and 9 geraniums for $90. Another landscaper bus 5 peonies and 6 geraniums for $85. Write and solve a sstem of linear equations to find the cost of each peon. Methods for Solving Sstems of Linear Equations Method 7 Chapter MSCC_ALG_PE_00.indd 7 When to Use Graphing (Lesson.) To estimate solutions Substitution (Lesson.) When one of the variables in one of the equations has a coefficient of or Elimination (Lesson.) When at least one pair of like terms has the same or opposite coefficients Elimination (Multipl First) (Lesson.) When one of the variables cannot be eliminated b adding or subtracting the equations Solving Sstems of Linear Equations Copright Big Ideas Learning, LLC /9/ :8:8 PM
22 . Eercises Help with Homework. WRITING Describe how to solve a sstem of linear equations b elimination.. NUMBER SENSE When should ou use multiplication to solve a sstem of linear equations b elimination?. WHICH ONE DOESN T BELONG? Which sstem of equations does not belong with the other three? Eplain our reasoning. + = = 7 + = 6 = 0 + = = 0 + = 5 = 9+(6)= +()= +(9)= 9+()= Use a method from Activit to solve the sstem.. + = 5. + = = = + = = 9 Solve the sstem of linear equations b elimination. Check our solution = 5 8. = = 5 = + = + = =. + 5 = 6. = = 5 = 6 =. ERROR ANALYSIS Describe and correct the error in solving the sstem of linear equations. 5 + = 9 Equation = Equation = 0 = 5. RAFFLE TICKETS You and our The solution is (5, 8). friend are selling raffle tickets for a new laptop. You sell more tickets than our friend sells. Together, ou and our friend sell 58 tickets. a. Write a sstem of linear equations that represents this situation. b. How man tickets do each of ou sell? 5. JOGGING You can jog around our block twice and the park once in 0 minutes. You can jog around our block twice and the park times in minutes. a. Write a sstem of linear equations that represents this situation. b. How long does it take ou to jog around the park? Copright Big Ideas Learning, LLC Section. Solving Sstems of Linear Equations b Elimination 7
23 Solve the sstem of linear equations b elimination. Check our solution. 6. = = 8. + = 7 = + 5 = = 9. + = 0. 6 =. 5 = = 7 = + 9 =. ERROR ANALYSIS Describe and correct the error in solving the sstem of linear equations. + = Equation Multipl b = = Equation 5 + = 8 = 8 = The solution is (, ).. REASONING For what values of a and b should ou solve the sstem b elimination? a. = b. 7 = 6 a + 0 = b = 9. AIRPLANES Two airplanes are fling to the same airport. Their positions are shown in the graph. Write a sstem of linear equations that represents this situation. Solve the sstem b elimination to justif our answer Airport TEST PRACTICE The table shows the number of correct answers on a practice standardized test. You score 86 points on the test and our friend scores 76 points. You Your Friend Multiple Choice 8 Short Response 0 5 a. Write a sstem of linear equations that represents this situation. b. How man points is each tpe of question worth? 7 Chapter Solving Sstems of Linear Equations Copright Big Ideas Learning, LLC
24 6. LOGIC You solve a sstem of equations in which represents the number of adult tickets sold and represents the number of student tickets sold. Can ( 6, ) be the solution of the sstem? Eplain our reasoning. 7. VACATION The table shows the activities of two tourists at a vacation resort. You want to go parasailing for one hour and horseback riding for two hours. How much do ou epect to pa? Parasailing Horseback Riding Total Cost Tourist hours 5 hours $05 Tourist hours hours $0 8. REASO O REASONING The solution of a sstem of linear equations is (, ) ). One equation in the sstem is + = 0. Eplain o could find a second equation for the sstem. Then how ou s find a second equation. Solve the sstem b elimination to our answer. justif 9 mi 9. JEWELER A metal allo is a m miture of two or more metals. A jeweler wants to make 8 grams of 8carat gold, which is 75% gold. The jeweler has an allo that is 90% gold and an allo that is 50% gold. How much of each allo should the jeweler use? 0. PROBLEM SOLVING A power boat takes 0 minutes to travel 0 miles downstream. The return trip takes 50 minutes. What is the speed of the current?. Solve the sstem of equations b elimination. + z = + z = z = 0 Decide whether the two equations are equivalent.. n + = n 8 (Section. and Section.). 7v = 5. a + 6 = n = 9 a+=6 v = 5 5. MULTIPLE CHOICE Which line has the same slope as =? A = + Copright Big Ideas Learning, LLC MSCC_ALG_PE_00.indd 75 B = + Section. C = 5 (Section.) D = 7 Solving Sstems of Linear Equations b Elimination 75 /9/ :8: PM
25 . Solving Special Sstems of Linear Equations Can a sstem of linear equations have no solution? Can a sstem of linear equations have man solutions? ACTIVITY: Writing a Sstem of Linear Equations Work with a partner. Your cousin is ears older than ou. Your ages can be represented b two linear equations. = t = t + Your age Your cousin s age a. Graph both equations in the same coordinate plane. b. What is the vertical distance between the two graphs? What does this distance represent? c. Do the two graphs intersect? If not, what does this mean in terms of our age and our cousin s age? t ACTIVITY: Using a Table to Solve a Sstem Work with a partner. You invest $500 for equipment to make dog backpacks. Each backpack costs ou $5 for materials. You sell each backpack for $5. COMMON CORE Sstems of Equations In this lesson, ou will solve sstems of linear equations having no solution or infinitel man solutions. Learning Standards 8.EE.8a 8.EE.8b 8.EE.8c A.CED. A.REI.6 a. Cop and complete the table for our cost C and our revenue R C R b. When will our compan break even? What is wrong? 76 Chapter Solving Sstems of Linear Equations Copright Big Ideas Learning, LLC
26 ACTIVITY: Using a Graph to Solve a Puzzle Math Practice Analze Relationships What do ou know about the graphs of the two equations? How does this relate to the number of solutions? Work with a partner. Let and be two numbers. Here are two clues about the values of and. Clue : Clue : Words is more than twice the value of. The difference of and 6 is. Equation = + 6 = a. Graph both equations in the same coordinate plane. b. Do the two lines intersect? Eplain c. What is the solution of the puzzle? d. Use the equation = + to complete the table e. Does each solution in the table satisf both clues? f. What can ou conclude? How man solutions does the puzzle have? How can ou describe them?. IN YOUR OWN WORDS Can a sstem of linear equations have no solution? Can a sstem of linear equations have man solutions? Give eamples to support our answers. Use what ou learned about special sstems of linear equations to complete Eercises and on page 80. Copright Big Ideas Learning, LLC Section. Solving Special Sstems of Linear Equations 77
27 Lesson. Lesson Tutorials Solutions of Sstems of Linear Equations A sstem of linear equations can have one solution, no solution, or infinitel man solutions. O EXAMPLE O No solution Infinitel man solutions The lines intersect. The lines are parallel. The lines are the same. Solving a Sstem: No Solution = + Equation = 5 Equation Graph each equation. The lines have the same slope and different intercepts. So, the lines are parallel. 5 = 5 = + 5 O Method : Solve b graphing. One solution Solve the sstem. 6 Because parallel lines do not intersect, there is no point that is a solution of both equations. So, the sstem of linear equations has no solution. Method : Solve b substitution. Substitute 5 for in Equation. = + Equation 5 = + 5 Substitute 5 for. Subtract from each side. The equation 5 = is never true. So, the sstem of linear equations has no solution. Solve the sstem of linear equations. Check our solution. Eercises 8 0. = +. = Chapter MSCC_ALG_PE_00.indd 78 Solving Sstems of Linear Equations = = 0. = = Copright Big Ideas Learning, LLC /9/ :0:0 PM
28 EXAMPLE Solving a Sstem: Infinitel Man Solutions The perimeter of the rectangle is 6 units. The perimeter of the triangle is 08 units. Write and solve a sstem of linear equations to find the values of and. Perimeter of rectangle Perimeter of triangle () + () = = = 6 Equation + = 08 Equation The sstem is: + 8 = 6 Equation + = 08 Equation Method : Solve b graphing. Graph each equation. The lines have the same slope and the same intercept. So, the lines are the same. Because the lines are the same, all the points on the line are solutions of both equations. So, the sstem of linear equations has infinitel man solutions Method : Solve b elimination. Multipl Equation b and subtract the equations. + 8 = 6 Multipl b. + = 08 Revised Equation + = 08 + = 08 Equation 0 = 0 Subtract. The equation 0 = 0 is alwas true. So, the solutions are all the points on the line + 8 = 6. The sstem of linear equations has infinitel man solutions. Eercises Solve the sstem of linear equations. Check our solution.. + = 5. + = 5 6. = 0 = + = 0 + = WHAT IF? What happens to the solution in Eample if the perimeter of the rectangle is 5 units? Eplain. Copright Big Ideas Learning, LLC Section. Solving Special Sstems of Linear Equations 79
29 . Eercises Help with Homework. WRITING Describe the difference between the graph of a sstem of linear equations that has no solution and the graph of a sstem of linear equations that has infinitel man solutions.. REASONING When solving a sstem of linear equations algebraicall, how do ou know when the sstem has no solution? infinitel man solutions? 9+(6)= +()= +(9)= 9+()= Let and be two numbers. Find the solution of the puzzle.. is more than times the value of.. of plus is equal to. The difference of and is. is 6 more than twice the value of. Without graphing, determine whether the sstem of linear equations has one solution, infinitel man solutions, or no solution. Eplain our reasoning. 5. = = = 8 = = + 8 = Solve the sstem of linear equations. Check our solution. 8. = 9. = + 0. = π + π = = π + = 6π. = =. + =. + 6 = = 6 (0.5 ) =.6. ERROR ANALYSIS Describe and correct the error in solving the sstem of linear equations. = + = + 6 The lines have the same slope so there are infinitel man solutions. 5. PIG RACE In a pig race, our pig gets a head start of feet and is running at a rate of feet per second. Your friend s pig is also running at a rate of feet per second. A sstem of linear equations that represents this situation is = + and =. Will our friend s pig catch up to our pig? Eplain. 80 Chapter Solving Sstems of Linear Equations Copright Big Ideas Learning, LLC
30 6. REASONING One equation in a sstem of linear equations has a slope of. The other equation has a slope of. How man solutions does the sstem have? Eplain. 7. LOGIC How can ou use the slopes and intercepts of equations in a sstem of linear equations to determine whether the sstem has one solution, infinitel man solutions, or no solution? Eplain our reasoning. + 8 = = 8 8. MONEY You and a friend both work two different jobs. The sstem of linear equations represents the total earnings for hours worked at the first job and hours worked at the second job. Your friend earns twice as much as ou. a. One week, both of ou work hours at the first job. How man hours do ou and our friend work at the second job? b. Both of ou work the same number of hours at the second job. Compare the number of hours ou each work at the first job. 9. DOWNLOADS You download a digital album for $0. Then ou and our friend download the same number of individual songs for $0.99 each. Write a sstem of linear equations that represents this situation. Will ou and our friend spend the same amount of mone? Eplain. 0. REASONING Does the sstem shown alwas, sometimes, or never have no solution when a = b? a b? a < b? Eplain our reasoning. = a + = b +. SKIING The table shows the number of lift tickets and ski rentals sold to two different groups. Is it possible to determine how much each lift ticket costs? Justif our answer.. Precision Find the values of a and b so the sstem shown has the solution (, ). Does the sstem have an other solutions? Eplain. Group Number of Lift Tickets 6 Number of Ski Rentals 8 Total Cost (dollars) b = a b = 6 Graph the inequalit in a coordinate plane. (Section.5) < 6. MULTIPLE CHOICE What is the solution of ( + 5) 6? (Section.) A B C D Copright Big Ideas Learning, LLC Section. Solving Special Sstems of Linear Equations 8
31 Etension. Solving Linear Equations b Graphing Lesson Tutorials COMMON CORE Sstems of Equations Solving Equations Using Graphs In this etension, ou will solve linear equations b graphing a sstem of linear equations. Learning Standards 8.EE.8a 8.EE.8b 8.EE.8c A.CED. A.REI.6 Step : To solve the equation a + b = c + d, write two linear equations. a + b = c + d = a + b Step : EXAMPLE = c + d and Graph the sstem of linear equations. The value of the solution of the sstem of linear equations is the solution of the equation a + b = c + d. Solving an Equation Using a Graph Solve = + using a graph. Check our solution. Step : Write a sstem of linear equations using each side of the equation. = + = + = Check Step : Graph the sstem. = +? = () + 0=0 = = (, 0) = + 5 The graphs intersect at (, 0). So, the solution of the equation is =. = + Use a graph to solve the equation. Check our solution.. + =. = + 8. =. + = = 6. = + 7. STRUCTURE Write an equation with variables on both sides that has no solution. How can ou change the equation so that it has infinitel man solutions? 8 Chapter MSCC_ALG_PE_00_et.indd 8 Solving Sstems of Linear Equations Copright Big Ideas Learning, LLC /7/ :8:6 PM
32 EXAMPLE RealLife Application Plant A Plant B Plant A grows 0.6 inch per month. Plant B grows twice as fast. in. 9 in. a. Use the model to write an equation. b. After how man months are the plants the same height? Growth rate Months, + Original height = Growth rate Months, + Original height a. The equation is = Stud Tip You can check our answer algebraicall as in Section =. + 9 = = = b. Write a sstem of linear equations using each side of the equation. Then use a graphing calculator to graph the sstem =. + 9 = = The solution of the sstem is (5, 5). So, the plants are both 5 inches tall after 5 months. Use a graph to solve the equation. Check our solution = + 9. = = WHAT IF? In Eample, the growth rate of Plant A is 0.5 inch per month. After how man months are the plants the same height? Copright Big Ideas Learning, LLC Etension. Solving Linear Equations b Graphing 8
33 .5 Sstems of Linear Inequalities sstem of linear inequalities? How can ou sketch the graph of a ACTIVITY: Graphing Linear Inequalities Work with a partner. Match the linear inequalit with its graph. + Inequalit 0 Inequalit a. b ACTIVITY: Graphing a Sstem of Linear Inequalities COMMON CORE Sstems of Inequalities In this lesson, ou will write and graph sstems of linear inequalities in two variables. solve reallife problems. Learning Standards A.CED. A.REI. Work with a partner. Consider the sstem of linear inequalities given in Activit. + Inequalit 0 Inequalit Use colored pencils to shade the solutions of the two linear inequalities. When ou graph both inequalities in the same coordinate plane, what do ou get? Describe each of the shaded regions in the graph at the right. What does the unshaded region represent? Chapter Solving Sstems of Linear Equations Copright Big Ideas Learning, LLC
34 ACTIVITY: Writing a Sstem of Linear Inequalities Work with a partner. Write a sstem of linear inequalities whose solution is the traffic sign at the right. () () + () 5 () ACTIVITY: Representing a State b a Linear Sstem Math Practice Use Clear Definitions What is a sstem of linear inequalities? How can this definition help ou identif the states? Two states can be represented as the graph of a sstem of linear inequalities. Identif the two states. Eplain our reasoning. 5. IN YOUR OWN WORDS How can ou sketch the graph of a sstem of linear inequalities? 6. When graphing a sstem of linear inequalities, which region represents the solution of the sstem? Do ou think all sstems have a solution? Eplain. Use what ou learned about sstems of linear inequalities to complete Eercises 7 9 on page 89. Copright Big Ideas Learning, LLC Section.5 Sstems of Linear Inequalities 85
35 .5 Lesson Lesson Tutorials Ke Vocabular A sstem of linear inequalities is a set of two or more linear inequalities in the same variables. An eample is shown below. sstem of linear inequalities, p. 86 solution of a sstem of linear inequalities, p. 86 graph of a sstem of linear inequalities, p. 86 < + Inequalit Inequalit A solution of a sstem of linear inequalities in two variables is an ordered pair that is a solution of each inequalit in the sstem. EXAMPLE Checking Solutions Tell whether each ordered pair is a solution of the sstem. Inequalit < + a. (, 5) Inequalit b. (, 0) Inequalit Inequalit Inequalit Inequalit <? 5 < () +? 5 + <? 0 < ( ) +? 0 + 5<6 5 0 < 0 (, 5) is a solution of both inequalities. So, it is a solution of the sstem. Eercises 6 (, 0) is not a solution of both inequalities. So, it is not a solution of the sstem. Tell whether the ordered pair is a solution of the sstem of linear inequalities < 5 > ; (, 5) < + ; (0, ) The graph of a sstem of linear inequalities is the graph of all of the solutions of the sstem. 6 5 Graphing a Sstem of Linear Inequalities Step Step 86 Chapter MSCC_ALG_PE_005.indd 86 Graph each inequalit in the same coordinate plane. Find the intersection of the halfplanes. This intersection is the graph of the sstem. Solving Sstems of Linear Equations à q Ź Ź Ź 5 Copright Big Ideas Learning, LLC /9/ :0: PM
36 EXAMPLE Graphing a Sstem of Linear Inequalities Stud Tip For help with graphing linear inequalities, see Section.5. Graph the sstem. Step : Graph each inequalit. Inequalit > + Inequalit Step : Find the intersection of the halfplanes. One solution is (, ). (, ) 5 Check Verif that (, ) is a solution of each inequalit. 5 Inequalit Inequalit > + >? + > The solution is the purple shaded region. EXAMPLE Graphing a Sstem of Linear Inequalities: No Solution Graph the sstem. + < Inequalit + > Inequalit Step : Graph each inequalit. Step : Find the intersection of the halfplanes. The lines are parallel and the halfplanes do not intersect. So, the sstem has no solution. Eercises 7 5 Graph the sstem of linear inequalities.. +. > 5. + < > Copright Big Ideas Learning, LLC Section.5 Sstems of Linear Inequalities 87
37 EXAMPLE Writing a Sstem of Linear Inequalities Write a sstem of linear inequalities represented b the graph. The horizontal boundar line passes through (0, ). So, an equation of the line is. The slope of the other boundar line is and the intercept is 0. So, an equation of the line is. The shaded region is above the solid boundar line, so the inequalit is. 5 The shaded region is below the dashed boundar line, so the inequalit is <. The sstem is and <. EXAMPLE 5 RealLife Application You have at most 8 hours to spend at the mall and at the beach. You want to spend at least hours at the mall and more than hours at the beach. Write and graph a sstem that represents the situation. How much time could ou spend at each location? Use the constraints to write a sstem of linear inequalities. Let be the number of hours at the mall and let be the number of hours at the beach. + 8 > at most 8 hours at the mall and at the beach at least hours at the mall more than hours at the beach Graph the sstem. One ordered pair in the solution region is (.5, 5). So, ou could spend.5 hours at the mall and 5 hours at the beach. Eercises 6 Write a sstem of linear inequalities represented b the graph WHAT IF? In Eample 5, ou want to spend at least hours at the mall. How does this change the sstem? Is (.5, 5) still a solution? Eplain. 88 Chapter Solving Sstems of Linear Equations Copright Big Ideas Learning, LLC
38 Eercises.5 Eercises Help with Homework. VOCABULARY How can ou verif that an ordered pair is a solution of a sstem of linear inequalities?. WRITING How are solving sstems of linear inequalities and sstems of linear equations similar? How are the different?. REASONING Is the point shown a solution of the sstem of linear inequalities? Eplain. Ź 5 Ź (, Ź) Ź Ź Ź5 6)= 9+( )= +( 9)= +( = ) 9+( Tell whether the ordered pair is a solution of the sstem of linear inequalities.. < > + ; ( 5, ) 5. > 5 ; (, ) ; (0, 0) Graph the sstem of linear inequalities. 7. < 5 0. < >. + < 8. > > <. + > < > < 5 > MUFFINS You can spend at most $ on fruit. Blueberries cost $ per pound and strawberries cost $ per pound. You need at least pounds to make muffins. a. Define the variables. alities b. Write a sstem of linear inequalities that represents this situation. c. Graph the sstem of linear inequalities. qualitie ie es. s. blueberrie iess d. Is it possible to bu pounds off blueberries and pound of strawberries in this situation? Justif our answer. Copright Big Ideas Learning, LLC MSCC_ALG_PE_005.indd 89 Section.5 Sstems of Linear Inequalities 89 /9/ :0:5 PM
39 ERROR ANALYSIS Describe and correct the error in graphing the sstem of linear inequalities < > Match the graph with the corresponding sstem of linear inequalities A. < + B. + C. + < + > + >. REASONING Describe the intersection of the halfplanes of the sstem shown.. JOBS You earn $ per hour working as a manager at a grocer store. You also coach a soccer team for $0 per hour. You need to earn at least $0 per week, but ou do not want to work more than 0 hours per week. a. Write and graph a sstem of linear inequalities that represents this situation. b. Identif and interpret one solution of the sstem. 90 Chapter Solving Sstems of Linear Equations Copright Big Ideas Learning, LLC
40 Write a sstem of linear inequalities represented b the graph Graph the sstem of linear inequalities. 7. > < > > < STRUCTURE Write a sstem of linear inequalities that is equivalent to < where > 0. Graph the sstem.. REPEATED REASONING One inequalit in a sstem is + > 6. Write another inequalit so the sstem has (a) no solution and (b) infinitel man solutions.. AMUSEMENT PARK You have at most 8 hours to spend at an amusement park. You want to spend less than hours plaing games and at least hours on rides. How much time can ou spend on each activit?. ROAD TRIP On a road trip, ou drive about 70 miles per hour and our friend drives about 60 miles per hour. The plan is to drive less than 5 hours and at least 600 miles each da. Your friend will drive more hours than ou. Identif and interpret one solution of this situation.. The following points are the vertices of a triangle. (, 5), (6, ), (, ) a. Write a sstem of linear inequalities that represents the triangle. b. Find the area of the triangle. Evaluate the epression when a =, b =, and c =. (Skills Review Handbook) 5. a bc 6. ab + c 7. c ac 8. MULTIPLE CHOICE What is the solution of ( ) = ( + )? (Section.) A = 5 B = C = 5 D = 7 Copright Big Ideas Learning, LLC Section.5 Sstems of Linear Inequalities 9
41 ..5 Quiz Progress Check Solve the sstem of linear equations b elimination. Check our solution. (Section.). + =. =. = + 0 = + = 0 + = Solve the sstem of linear equations. Check our solution. (Section.). = 6 5. = = 6 = + = + = Use a graph to solve the equation. Check our solution. (Section.) 7. = 8. + = + 9. = + Graph the sstem of linear inequalities. (Section.5) > 0 > < Write a sstem of linear inequalities represented b the graph. (Section.5) RENTALS A business rents biccles and inline skates. Biccle rentals cost $5 per da and inline skate rentals cost $0 per da. The business has 0 rentals toda and makes $55. (Section.) a. Write a sstem of linear equations that represents this situation. b. How man biccle rentals and inline skate rentals did the business have toda? 6. JOBS You earn $ per hour delivering pizzas. You also work parttime at a convenience store where ou earn $9 per hour. You want to earn at least $50 per week, but ou can onl work 5 hours per week. How man hours can ou work at each job? (Section.5) 9 Chapter Solving Sstems of Linear Equations Copright Big Ideas Learning, LLC
42 Chapter Review Review Ke Vocabular sstem of linear equations, p. 56 solution of a sstem of linear equations, p. 56 sstem of linear inequalities, p. 86 solution of a sstem of linear inequalities, p. 86 Vocabular Help graph of a sstem of linear inequalities, p. 86 Review Eamples and Eercises. Solving Sstems of Linear Equations b Graphing (pp. 5 59) Solve the sstem b graphing. = Equation = + 5 Equation Step : Graph each equation. Step : Estimate the point of intersection. The graphs appear to intersect at (, ). Step : Check the point from Step. = = + 5 =? ( ) =? ( ) + 5 = = The solution is (, ). 6 5 (, ) 5 Solve the sstem of linear equations b graphing.. =. = +. = = + + = 0 =. Solving Sstems of Linear Equations b Substitution (pp ) Solve the sstem b substitution. = + Equation + = Equation Step : Equation is alread solved for. Step : Substitute + for in Equation. + + = Substitute + for. = Solve for. Step : Substituting for in Equation gives =. The solution is (, ). Copright Big Ideas Learning, LLC Chapter Review 9
43 Solve the sstem of linear equations b substitution. Check our solution.. = = = 8 = + 9 = = 0. Solving Sstems of Linear Equations b Elimination (pp ) You have a total of 5 quarters and dimes in our pocket. The value of the coins is $0.80. Write and solve a sstem of linear equations to find the number of dimes and the number of quarters in our pocket. Use a verbal model to write a sstem of linear equations. Value of a dime Number of dimes, Number of dimes, Value of a + quarter The sstem is + = 5 and = 0.8. Step : Multipl Equation b 0. + Number of quarters, = Number of coins Number of quarters, = Total value + = 5 + = 5 Equation = 0.8 Multipl b = 8 Revised Equation Step : Subtract the equations. + = 5 Equation +.5 = 8 Revised Equation.5 = Subtract the equations. Step : Solving the equation.5 = gives =. Step : Substitute for in one of the original equations and solve for. + = 5 Equation + = 5 Substitute for. = Subtract from each side. So, ou have dimes and quarters in our pocket. 7. GIFT BASKET A gift basket that contains jars of jam and packages of bread mi costs $5. There are 8 items in the basket. Jars of jam cost $6 each and packages of bread mi cost $5 each. Write and solve a sstem of linear equations to find the number of jars of jam and the number of packages of bread mi in the gift basket. 9 Chapter Solving Sstems of Linear Equations Copright Big Ideas Learning, LLC
44 . Solving Special Sstems of Linear Equations (pp. 76 8) Solve the sstem. = 5 8 Equation = 5 + Equation Solve b substitution. Substitute 5 + for in Equation. = 5 8 Equation 5 + = 5 8 Substitute 5 + for. 8 Add 5 to each side. The equation = 8 is never true. So, the sstem of linear equations has no solution. Solve the sstem of linear equations. Check our solution = 5 9. = 0. 8 = 6 = = + = 8. Use a graph to solve 9 = 7 +. Check our solution..5 Sstems of Linear Inequalities (pp. 8 9) Graph the sstem. < Inequalit Inequalit Check Step : Graph each inequalit. Verif that (0, ) is a solution of each inequalit. Inequalit Inequalit < Step : Find the intersection of the halfplanes. One solution is (0, ). 5 (0, ) <? 0? (0) < 7 The solution is the purple shaded region. Graph the sstem of linear inequalities... > +. + > + + < Copright Big Ideas Learning, LLC Chapter Review 95
45 Chapter Test Solve the sstem of linear equations b graphing. Test Practice. =. = = 0 = = + 6 = 9 Solve the sstem of linear equations b substitution. Check our solution.. + = 5. + = 0 6. = + = 0 = + = 6 Solve the sstem of linear equations b elimination. Check our solution = 8. + = = 0 = = = Without graphing, determine whether the sstem of linear equations has one solution, infinitel man solutions, or no solution. Eplain our reasoning. 0. = + 8. = 6. = + = 5 + = = 0 Use a graph to solve the equation. Check our solution.. = +. 8 = Graph the sstem of linear inequalities. 5. > < + + > 5 + > 8. BOUQUET A bouquet of lilies and tulips has flowers. Lilies cost $ each and tulips cost $ each. The bouquet costs $. Write and solve a sstem of linear equations to find the number of lilies and tulips in the bouquet. 9. DINNER How much does it cost for two specials and two glasses of milk? 0. SHOPPING You have $0 to spend at the mall. You want to bu at most 6 articles of clothing. A clothing store sells shirts for $ and pairs of pants for $8. You want to have at least $0 left over for food. a. Write and graph a sstem of linear inequalities that represents this situation. b. How man shirts and pairs of pants can ou bu at the store? 96 Chapter Solving Sstems of Linear Equations Copright Big Ideas Learning, LLC
46 Standards Assessment. What is the solution of the sstem of equations shown below? (A.REI.6) TestTaking Strateg Read Question Before Answering = + 6 = 6 A. (, 0 ) C. No solution B. (0, ) D. Infinitel man solutions. Which inequalit is shown in the coordinate plane? (A.REI.) Take our time and read the question carefull before choosing our answer. F. > 5 G. < 5 H. 5 I What is the slope of a line that is perpendicular to the line = 0.5 +? (F.IF.6). Which graph shows the solution of + >? (A.REI.) A. C. B. D. Copright Big Ideas Learning, LLC Standards Assessment 97
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