Graphically Solving Linear Systems. Matt s health-food store sells roasted almonds for $15/kg and dried cranberries for $10/kg.

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1 1.3 Graphicall Solving Linear Sstems GOAL Use graphs to solve a pair of linear equations simultaneousl. INVESTIGATE the Math Matt s health-food store sells roasted almonds for $15/kg and dried cranberries for $1/kg.? How can he mi the almonds and the cranberries to create 1 kg of a miture that he can sell for $1/kg? YOU WILL NEED grid paper ruler graphing calculator A. Let represent the mass of the almonds. Let represent the mass of the cranberries. i) Write an equation for the total mass of the miture. ii) Write an equation for the total value of the miture. B. Graph our equation of the total mass for part A. What do the points on the line represent? C. Graph our equation of the total value for part A on the same aes. What do the points on this line represent? D. Identif the coordinates of the point where the two lines intersect. State what each value represents. How accuratel can ou estimate these values from our graph? E. The equations for part A form a sstem of linear equations. Eplain wh the coordinates for part D give the solution to a sstem of linear equations. F. Substitute the coordinates into each equation to verif our solution. Reflecting G. Eplain wh ou needed two linear relations to describe the problem. H. Eplain how graphing both relations on the same aes helped ou solve the problem. I. Eplain wh the coordinates of the point of intersection provide an ordered pair that satisfies both relations. sstem of linear equations a set of two or more linear equations with two or more variables For eample, 1 solution to a sstem of linear equations the values of the variables in the sstem that satisf all the equations For eample, (7, 3) is the solution to 1 NEL Chapter 1 1

2 APPLY the Math EXAMPLE 1 Selecting a graphing strateg to solve a linear sstem Solve the sstem = + 1 and + = -8 using a graph. Leslie s Solution The slope of the line is. 1 = rise run = + 1 At the -intercept,. () = -8 At the -intercept, At the -intercept, =. =. + () = -8 + = -8 = -8 = - 8 = At the point of intersection, = - and = -3. The solution is (, 3) I determined the slope and the -intercept of the first equation. I determined the - and -intercepts of the second equation. I graphed the first line (blue) b plotting the -intercept and using the rise and run to plot another point on the line. I graphed the second line (red) b plotting points ( 8, ) and (, ) and joining them with a straight line. I located the point of intersection and read its coordinates using the aes of m graph. 1 Left Side Right Side 1 3 ( ) Left Side ( 3) 8 Right Side 8 I checked m solution b substituting the - and -values into each equation. 1.3 Graphicall Solving Linear Sstems NEL

3 1.3 EXAMPLE Solving a problem using a graphing strateg Ellen drives 5 km from her universit in Kitchener-Waterloo to her home in Smiths Falls. She travels along one highwa to Kingston at 1 km/h and then along another highwa to Smiths Falls at 8 km/h. The journe takes her h 5 min. What is the distance from Kingston to Smiths Falls? Bob s Solution Let represent the distance that Ellen travels at 1 km/h. Let represent the distance that she travels at 8 km/h. The total trip is 5 km, so = 3 + = 5 At the -intercept,. At the -intercept, = 3 At the -intercept,. At the -intercept,. 1 + = = 3 = 1a 3 b = 1a 19 b = 75 = 8a 3 b = 8a 19 b = 38 I used letters to identif the variables in this situation. I wrote an equation for the total distance. Since speed = distance, then time = distance time. speed I wrote an equation to describe the total time (in hours) for her trip, where 1 is the time spent driving at 1 km/h and is the time spent driving 8 at 8 km/h. I determined the - and -intercepts of the first equation. I determined the - and -intercepts of the second equation. NEL Chapter 1 3

4 Distance at 8 km/h (km) Distance at 8 km/h vs. Distance at 1 km/h Distance at 1 km/h (km) I graphed each equation b plotting the - and -intercepts and joining them with a straight line. The point of intersection is (35, 1), so the distance from Kingston to Smiths Falls is 1 km. I determined the coordinates of the point of intersection. The -coordinate of the point is the distance. EXAMPLE 3 Selecting graphing technolog to solve a sstem of linear equations Hale wants to rent a car for a weekend trip. Kell s Kars charges $95 for the weekend plus $.15/km. Rick s Rentals charges $5 for the weekend plus $.6/km. Which compan charges less? Ell s Solution Let represent the distance driven in kilometres. Let represent the total cost of the car rental in dollars. The cost to rent a car from Kell s Kars is The cost to rent a car from Rick s Rentals is.6 5. I chose and for the variables. I wrote an equation for the cost to rent a car from Kell s Kars. $.15 represents the distance charge and $95 represents the weekend fee. I wrote an equation for the cost to rent a car from Rick s Rentals. $.6 represents the distance charge and $5 represents the weekend fee. 1.3 Graphicall Solving Linear Sstems NEL

5 1.3 I used a graphing calculator. I entered the equation for Kell s Kars in Y1 and the equation for Rick s Rentals in Y. I used a thick line for Rick s Rentals to distinguish between the two lines. Tech Support To graph with a thick line using a TI-83/8 graphing calculator, scroll to the left of Y and press. ENTER The point of intersection is (9, 156), to the nearest whole numbers. If Hale drives 9 km, both companies charge about $156. If she plans to drive less than 9 km, Rick s Rentals charges less. If she plans to drive more than 9 km, Kell s Kars charges less. I graphed the lines and adjusted the window settings so that I could see both lines and the point of intersection. The point of intersection occurred when X 6 and Y. I used the Intersect operation to determine the point of intersection. I looked at m graph to determine which line is lower before and after the point of intersection. Before the point of intersection, the thick line is lower, so Rick s Rentals charges less. After the point of intersection, the thin line is lower, so Kell s Kars charges less. Tech Support For help using a TI-83/8 graphing calculator to determine the point of intersection, see Appendi B-11. If ou are using a TI-nspire, see Appendi B-7. In Summar Ke Idea You can solve a sstem of linear equations b graphing both equations on the same aes. The ordered pair (, ) at the point of intersection gives the solution to the sstem. Need to Know You ma not be able to determine an accurate solution to a sstem of equations using a hand-drawn graph. To determine an accurate solution to a sstem of equations, ou can use graphing technolog. When ou use a graphing calculator, epress the equations in slope -intercept form. NEL Chapter 1 5

6 CHECK Your Understanding 1. Decide whether each ordered pair is a solution to the given sstem of equations. a) (, 1); 3 and 3 5 b) (1, ); 5 and 8 c) (1, ); 3 5 and d) (1, 5); 5 and 5. For each graph: i) Identif the point of intersection. ii) Verif our answer b substituting into the equations. a) b) 8 c) PRACTISING 3. a) Graph the sstem 5 and 3 1 b hand. K b) Solve the sstem using our graph. c) Verif our solution using graphing technolog.. Ale needs to rent a minivan for a week to take his band on tour. Easvans charges $3 plus $.1/km. Cars for All Seasons charges $15 plus $./km. a) Write an equation for each rental compan. b) Graph our equations. c) Which rental compan would ou recommend to Ale? Eplain. 5. Solve each linear sstem b graphing. a) 3 d) 1 7 b) 8 e) c) f) Graphicall Solving Linear Sstems NEL

7 6. Austin is creating a new trail mi b combining two of his best-selling blends: a pineapple coconut macadamia mi that sells for $18/kg and a banana papaa peanut mi that sells for $1/kg. He is making 8 kg of the new mi and will sell it for $1.5/kg. Austin uses the graph shown at the right to determine how much of each blend he needs to use. a) Write the equations of the linear relations in the graph. b) From the graph, how much of each blend will Austin use? 7. At Jessica s Java, a new blend of coffee is featured each week. This week, Jessica is creating a low-caffeine espresso blend from Brazilian and Ethiopian beans. She wants to make kg of this blend and sell it for $15/kg. The Brazilian beans sell for $1/kg, and the Ethiopian beans sell for $17/kg. How man kilograms of each kind of bean must Jessica use to make kg of her new blend of the week? 8. When Arthur goes fishing, he drives 393 km from his home in Ottawa to a lodge near Temagami. He travels at an average speed of 7 km/h along the highwa to North Ba and then at 5 km/h on the narrow road from North Ba to Temagami. The journe takes him 6 h. a) Write two equations to describe this situation. b) Graph our equations. c) Use our graph to determine the distance from North Ba to Temagami. 9. Joanna is considering two job offers. Phoeni Fashions offers $15/month plus.5% commission. Stles b Rebecca offers $15/month plus 5.5% commission. a) Create a linear sstem b writing an equation for each salar. b) What value of sales would result in the same total salar for both jobs? c) Which job should Joanna take? Eplain our answer. 1. Create a situation ou can represent b a sstem of linear equations T that has the ordered pair (1, 15) as its solution. 11. Si cups of coffee and a dozen muffins originall cost $ The price of a cup of coffee increases b 1%. The price of a dozen muffins increases b 1%. The new cost of si cups of coffee and a dozen muffins is $17.6. Determine the new price of one cup of coffee and a dozen muffins. 1. Willow bought 3 m of denim fabric and 5 m of cotton fabric. The total bill, ecluding ta, was $. Jared bought 6 m of denim fabric and m of cotton fabric at the same store for $8. a) Write a linear sstem ou can solve to determine the price of denim fabric and the price of cotton fabric. b) Solve our sstem using a graph. c) How much will 8 m of denim fabric and 5 m of cotton fabric cost? Mass of banana mi (kg) 1.3 Trail Mi Mass of pineapple mi (kg) NEL Chapter 1 7

8 Environment Connection Fluorescent light bulbs decrease energ use and reduce pollution levels. 13. The drama department of a school sold 679 tickets to the school pla, for a total of $337. Students paid $ for a ticket, and non-students paid $7. a) Write a linear sstem for this situation. b) How man non-students attended the pla? Solve the problem b graphing our sstem. 1. The equations,, and 1 form the sides A of a triangle. a) Graph the triangle, and determine the coordinates of the vertices. b) Calculate the area of the triangle. 15. A regular light bulb costs $.65 to bu, plus $./h for the C electricit to make it work. A fluorescent light bulb costs $3.99 to bu, plus $.1/h for the electricit. a) Write a cost equation for each tpe of light bulb. b) Graph the sstem of equations using a graphing calculator. Use the window settings X and Y 1. c) After how long is the fluorescent light bulb cheaper than the regular light bulb? d) Determine the difference in cost after one ear of constant use. 16. Rearrange the following sentences to describe the correct sequence of steps for solving a problem b graphing a linear sstem. Discard an sentences that do not belong in the description. Add an sentences that are needed to make the description clearer. Label the graph. Verif the solution b substituting into both equations. Write two equations that describe the situation in the problem. Determine the slope of each line b calculating the rise over the run. Read the problem, and determine what ou need to find. Graph both equations on the same set of aes. Choose the best strateg to graph each equation. Determine the coordinates of the point of intersection. Etending 17. a) Solve the linear sstem 3 11 and 1. b) Show that the line with the equation 9 19 passes through the point where the lines in part a) intersect. c) Determine the values of c and d if 9 19 is written in the form c(3 11) d( 1). 18. Solve the linear sstem 1, 3 7, and Solve each sstem of equations. a) = b) = = = Graphicall Solving Linear Sstems NEL