8.3. Number of Solutions for Systems of Linear Equations. Investigate Number of Solutions for Systems of Linear Equations.

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1 8.3 Number of Solutions for Sstems of Linear Equations Focus on explaining wh sstems of linear equations can have different numbers of solutions identifing how man solutions a sstem of linear equations has solving problems involving linear sstems with different numbers of solutions Web Link To learn more about the Arctic Winter Games, go to and follow the links. At the Arctic Winter Games, Northern outh share cultural experiences and compete in various events. One of the events is snowshoe racing. Situations involving time and distance, such as a race, can be represented with a sstem of linear equations. If ou solve a linear sstem involving a race, will ou alwas expect a single solution? Materials stopwatch or other timer or clock showing seconds measuring tape grid paper and coloured pencils, or computer with graphing software Investigate Number of Solutions for Sstems of Linear Equations Work in groups of four to act out different race scenarios. Each group needs to assign two people to race, one person to measure time, and one person to record data. 1. As a group, design a data table to record start and end times and distances for each racer. 446 MHR Chapter 8

2 . Complete several different race scenarios. Each student should move at a constant speed for the entire race. For each race, record the data. Consider the starting line distance zero. 3. Create a distance-time graph to represent each race. You ma wish to use a different colour for each line. 4. Write a sstem of linear equations to represent each race. 5. Reflect and Respond Solve each sstem of linear equations ou graphed. How man solutions were there for each linear sstem? Explain how each solution relates to details of the race. 6. How could ou have predicted the number of solutions for each linear sstem just b knowing the starting points and speeds of the racers? Link the Ideas When two lines are graphed on the same grid, the do not alwas have exactl one point of intersection as seen in sections 8.1 and 8.. Parallel lines do not intersect at all. So, a sstem of parallel lines has no solution. Coincident lines have an infinite number of solutions because the lines are equivalent. The overlap. Parallel Lines 6 Coincident Lines 6 coincident lines lines that occup the same position in a graph of two coincident lines, an point of either line lies on the other line x x Number of Solutions for Sstems of Linear Equations MHR 447

3 Reducing the equations to lowest terms ma help ou identif whether the equations are equivalent. If the are equivalent, then the must have an infinite number of solutions. For the linear sstem x = 0 and 3x = 0, the first equation is in lowest terms, but the second equation is not. Dividing each side of the equation 3x = 0 b 3 gives x = 0, which is equivalent to the first equation. Therefore, the linear sstem has an infinite number of solutions. Example 1 Connect the Number of Solutions to the Situation Did You Know? Man breeds of dogs are used for competitive racing. Teams of Canadian Eskimo Dogs have helped people survive in the harsh climate of northern Canada for centuries. Canadian Eskimo Dogs help move people and materials throughout regions that are covered in snow during most of the winter. The Canadian Eskimo Dog is Nunavut s official mammal in honour of its vital role. A particular dog-mushing race is 13 km long. The distances and speeds for several competitors at a certain time during the race are shown in the table of values. Current Distance Travelled (km) Current Speed (km/h) Competitor A Competitor B Competitor C Competitor D Assume the racers continue at their current speeds. For each pair of competitors below, write a sstem of linear equations representing their travel from this point forward graph each sstem of linear equations identif and interpret the solution to each linear sstem a) competitors A and B b) competitors A and C c) competitors C and D 448 MHR Chapter 8

4 Solution Let d represent distance from the start of the race. Let t represent time from this point on in the race. a) Competitor A is travelling at 4 km/h and has travelled 6.0 km. Competitor B is travelling at 3 km/h and has travelled 5.0 km. Their travel can be represented b the following equations: d = 4t + 6 d = 3t + 5 The lines intersect at one point. So, the linear sstem has one solution. Competitor B will catch up to competitor A after 0.15 h, or 7.5 min, and then proceed past. The point (0.15, 9) is the onl point that lies on both lines. b) Competitors A and C are both travelling at 4 km/h. Competitor A has travelled 6.0 km. Competitor C has travelled 4.0 km. Their travel can be represented b the equations d = 4t + 6 and d = 4t + 4. The lines have the same slope but different -intercepts. The lines are parallel. The have no points in common, so the linear sstem has no solutions. Competitors A and C travel at the same speed, but at different distances from the start line. Competitor C will never catch up to competitor A. There is no point where the are at the same distance at the same time. 8.3 Number of Solutions for Sstems of Linear Equations MHR 449

5 c) Competitor C is travelling at 4 km/h and has travelled 4.0 km. Competitor D is also travelling at 4 km/h and has also travelled 4 km. The are currentl at the same distance and travelling at the same speed. Their travel can be represented b the equations d = 4t + 4 and d = 4t + 4. The lines have the same slope and the same -intercept. The lines are coincident, so the share all the same points. The linear sstem has an infinite number of solutions. Competitors C and D are side b side on the course. The will continue this wa because the are travelling at the same speed. Your Turn Four vehicles travel on a long, straight stretch of the Trans-Canada Highwa. Their current distances and speeds are shown in the table of values. Current Distance (km) Current Speed (km/h) Car Minivan 5 90 Truck RV For each pair of vehicles, represent the distance-time relationship using a sstem of linear equations. Suppose the vehicles continue at their current speeds. Identif and interpret the solution to each linear sstem. a) the car and the minivan b) the car and the RV c) the truck and the RV 450 MHR Chapter 8

6 Example Predict and Confirm the Number of Solutions Predict the number of solutions for each sstem of linear equations. Explain our reasoning, and then confirm each answer b graphing the linear sstem. a) = x - 3 b) 4x + 10 = 30 c) 10x - 6 = -1 = 1_ x + 3 x + 5 = 35 1 = x Solution a) The slope of = x - 3 is. The slope of = 1_ 8 x + 3 is 1_. 6 4 = _ 1 x + 3 = x x - The equations have different slopes. So, the graph will result in two lines that intersect at one point. Therefore, this sstem has one solution. b) Rearrange each equation to slope-intercept form b isolating. 4x + 10 = 30 x + 5 = 35 4x x = 30-4x x x = 35 - x 10 = -4x = -x + 35 = _- 5 x + 3 = _- 5 x + 7 Since the lines have the same slope and different -intercepts, the graph will result in parallel lines. The lines will never intersect. Therefore, this linear sstem has no solutions. 6 = - x = - x x 8.3 Number of Solutions for Sstems of Linear Equations MHR 451

7 c) For the linear sstem 10x - 6 = -1 and 1 = x, isolate in each equation to compare the slopes and -intercepts. 10x - 6 = -1 10x = x + 1 = 6 5_ 3 x + = = 5_ 3 x + 1 = x _ 1 1 = 4_ 1 + _ 35x 1 = + 5_ 3 x = 5_ 3 x + Both equations have a slope of 5_ and a -intercept of. 3 The graph will result in coincident lines. Therefore, this linear sstem has an infinite number of solutions x - Your Turn Predict the number of solutions for each sstem of linear equations. Justif our answers using a graph. a) x + = 4 b) 6-4x = 6 c) = 3x - 1 = - 1_ x + 4 = _ 3 x + 1 = x MHR Chapter 8

8 Example 3 Identif Zero and Infinite Solutions b Comparing Coefficients Sabrina s teacher gives her the following sstems of linear equations and tells her that each sstem has either no solution or an infinite number of solutions. How can Sabrina determine each answer b inspecting the equations? a) x + 3 = 1 b) x + 3 = 1 x + 3 = 0 4x + 6 = 4 Solution a) Sabrina notices that the left sides of the equations are identical. So, an ordered pair she substitutes will result in the same value on the left side of each equation. However, How else could ou confirm that the lines are parallel? the right sides are not equal. There are no ordered pairs that can satisf both equations, so the lines never intersect. Sabrina concludes that the linear sstem has no solutions. b) Sabrina notices that the second equation, 4x + 6 = 4, is not in lowest terms. She divides each term b. This results in an equation that is identical to the first equation, x + 3 = 1. Therefore, the equations are equivalent and the graph will be a pair of coincident lines. The linear sstem has an infinite number of solutions. _ 4x _ + 6 _ = 4 x + 3 = 1 Your Turn Determine, b inspection, whether each linear sstem has an infinite number of solutions or no solution. Explain our reasoning. a) x = 0 x = 0 b) x = 0 x = Number of Solutions for Sstems of Linear Equations MHR 453

9 Ke Ideas A sstem of linear equations can have one solution, no solution, or an infinite number of solutions. Before solving, ou can predict the number of solutions for a linear sstem b comparing the slopes and -intercepts of the equations. Intersecting Lines Parallel Lines Coincident Lines one solution no solution an infinite number of solutions - 0 x x - 0 x different slopes same slope same slope -intercepts can be different -intercepts same -intercept the same or different. For some linear sstems, reducing the equations to lowest terms and comparing the coefficients of the x-terms, -terms, and constants ma help ou predict the number of solutions. Check Your Understanding Practise 1. Which pair(s) of lines in the graph form a linear sstem that has a) exactl one solution? b) no solution? A 4 B C x D MHR Chapter 8

10 . Predict the number of solutions for each sstem of linear equations. Justif our answers. a) = x + b) = x - 4 c) = 3x + = x + = x + 1 = 3x How man solutions does each linear sstem have? Justif our answers. a) x + 3 = 6 b) 3x - = 1 c) x - 4 = 8 = - 1_ x + 6 4x - = 1 x + 4 = Describe the graph and the equations of a linear sstem that has a) no solution b) one solution c) an infinite number of solutions 5. Describe a strateg for predicting how man solutions each sstem of linear equations has just b looking at it. Test our strateg b graphing each linear sstem. a) x + 7 = 8 b) x + = 1 x + 7 = 15 x + 4 = 4 Appl 6. One of the equations in a linear sstem is x = 0. What might the other equation be if the sstem has a) no solutions? b) one solution? c) infinitel man solutions? 7. A provincial magazine emplos sales people who earn mone for ever subscription the sell. Several emploees are comparing their earnings so far this month. Current Earnings ($) Earnings Per Subscription ($) Alssa Brian Charlie Dena Write a sstem of equations to represent the earnings for each pair of emploees. Identif the solution to each sstem. Explain how the solution relates to the emploees earnings. a) Brian and Charlie b) Alssa and Brian c) Charlie and Dena 8.3 Number of Solutions for Sstems of Linear Equations MHR 455

11 8 x + 11_ and = 7_ 7 11 x + 8_ 5 using technolog. The use dimensions of -10 to 10 on the x-axis and -10 to 10 on the -axis. Sal thinks there is one solution. Jeff claims that there are an infinite number of solutions. 8. Sal and Jeff graph the sstem = 5_ a) Describe how to predict the number of solutions. b) Use technolog to recreate their graph. Explain wh each person might have come to his conclusion. 9. Stephanie graphs a sstem of equations using technolog. She obtains the following graph and concludes that the sstem has no solutions. a) Explain wh Stephanie might have made her conclusion. b) Is Stephanie correct? Explain. Did You Know? The Grouse Grind is the most used hiking trail in the Vancouver area. The trail is.9 km long and rises 853 m. Over people walk or run this steep climb ever ear. Maintenance workers have built steps up most of the trail to protect against erosion and for the safet of the hikers. 10. Alex, Sandra, Christine, and Jared are hiking the Grouse Grind near Vancouver, BC. Alex starts first. Sandra and Christine start a short time later. The hike together at a faster speed than Alex s. Jared starts last and hikes at the same speed as Alex. Each person travels at a constant rate. a) Sketch a graph that shows each person s progress up the mountain. b) Suppose sstems of linear equations are created to represent the travel for each possible pair of hikers. How man solutions would each sstem have? Explain how the number of solutions relates to the hike. 456 MHR Chapter 8

12 11. Is the following statement true or false? Explain and provide an example to support our answer. If a sstem of linear equations has an infinite number of solutions, then an pair of numbers is a solution to the sstem. 1. Suppose ou are given onl the following pieces of information about a sstem of linear equations. Would ou be able to predict the number of solutions to the sstem? Explain. a) The slopes of the lines are the same. b) The -intercepts of the lines are the same. c) The x-intercepts are the same, and the -intercepts are the same. 13. PaperWest has produced 5000 kg of napkins. It continues to manufacture 350 kg of napkins per week. Northern Paper manufactures napkins at a rate of 1400 kg per month and has alread produced kg. Assume one month has exactl four weeks. a) Write a sstem of linear equations to represent the manufacturing of the napkins. b) Explain how the number of solutions to the sstem relates to this situation. Extend 14. For the linear sstem x + 3 = 1 and 4x + 6 = C, what value(s) of C will give the sstem a) an infinite number of solutions? b) no solution? 15. Two taxis travel the same route from the airport. One taxi is 6 km from the airport and has a fuel econom of 0 km/l. The other taxi is just leaving the airport and uses 5 L of fuel for ever 100 km travelled. a) Create a sstem of linear equations relating the distance travelled ( kilometres) to the amount of fuel used (x litres) for each taxi. b) Explain how the number of solutions to the sstem relates to the travel of the taxis. 8.3 Number of Solutions for Sstems of Linear Equations MHR 457

13 16. Consider the sstem = 56 - x and = 10 + x. a) Suppose the domain for the sstem is restricted to 0 x 8. How man solutions does the sstem have? Explain. b) Suppose there are no restrictions on the domain. How man solutions does the sstem have? c) What effects do restrictions on the domain of the equations have when ou are predicting the number of solutions? Explain. 17. Consider the sstem Wx + 3 = W and 1x + W = 4. a) What value(s) of W give the sstem an infinite number of solutions? b) How man solutions will the sstem have for value(s) of W other than those ou found in part a)? Justif our answer. Create Connections 18. Two plants are growing according to the equations h = t and h = 55 + t. In the equations, h represents height, in centimetres, and t represents time, in weeks. Wend reasons that one plant is shorter and is growing slower, so it will never catch up to the other plant. Wend concludes that the sstem has no solutions. Harriet states that the two lines have different slopes, so there must be an intersection point. Who do ou agree with? Create a graph and use it to explain our answer. 19. Can a sstem of two linear equations have exactl two solutions? Explain our answer, using words and diagrams. 0. Do ou think it is alwas possible to tell how man solutions a sstem of linear equations has just b looking at the equations? Explain our thinking. 1. Describe a real-life situation that could be modelled b a sstem of linear equations having each number of solutions. Include examples of possible sstems of linear equations. a) no solution b) one solution c) an infinite number of solutions 458 MHR Chapter 8

14 . MINI LAB Investigate relationships between equations in a linear sstem and the number of solutions to the sstem. Step 1 Step Step 3 Step 4 Consider x + = 4 as equation #1. Create four linear sstems (A, B, C, and D) b writing equation # according to each of the following instructions. Sstem A: Multipl (both sides of) equation #1 b a number of our choice. Sstem B: Add a number of our choice to onl the right side of equation #1. Sstem C: Perform an operation of our choice on both sides of equation #1. Sstem D: Perform an operation of our choice on onl one side of equation #1. Use a graph to analse the number of solutions for each sstem of linear equations. What can ou conclude about the number of solutions for a sstem of linear equations if one equation is a multiple of the other? What if two different equations in general form have identical coefficients of x-terms and identical coefficients of -terms? Create an equation # so that the linear sstem has exactl one solution. Experiment with technolog to develop several different possibilities. What conclusions can ou make about the equations in a linear sstem that has exactl one solution? Materials graphing calculator or computer with graphing software 8.3 Number of Solutions for Sstems of Linear Equations MHR 459