CHAPTER 9 Linear Relations GET READY 474 Math Link 476 9.1 Warm Up 477 9.1 Analysing Graphs of Linear Relations 478 9.2 Warm Up 487 9.2 Patterns in a Table of Values 488 9.3 Warm Up 498 9.3 Linear Relationships 499 Chapter Review 507 Practice Test 512 Wrap It Up! 515 Key Word Builder 516 Math Games 517 Challenge in Real Life 518 Answers 519 Name: _____________________________________________________ Date: ______________ Describe Patterns in Words To describe a pattern in words, say: • what it is (number, letter, shape) • where it starts • how it changes (skips, decreases) Pattern 6, 9, 12, 15, … Description in Words • whole numbers • begins with 6 • increases by 3 each time 1. Describe each pattern in words. a) b, e, h, … b) 9, 4, –1, … • _______________________________ • _______________________________ • _______________________________ • _______________________________ • _______________________________ • _______________________________ Show Patterns in a Table A café has small tables that seat 4 people. Two tables put together seat 6 people. Three tables put together seat 8 people. Describe the pattern using a table of values. 1 4 Number of Tables Number of People 2 6 Describe the pattern using words: • The number of people begins at 4 and increases by 2 each time a table is added. 3 8 2. Complete the table of values and describe each pattern in words. a) Figure 1 Figure 2 b) Figure 3 Figure 3 Figure 2 Figure 1 Figure Number 1 2 3 Number of Squares 474 Figure Number 1 2 3 Number of Cubes Describe in words: Describe in words: ____________________________________ _________________________________ ____________________________________ _________________________________ MHR ● Chapter 9: Linear Relations Name: _____________________________________________________ Date: ______________ Describe Patterns Using an Expression expression ● a number or letter or combination of numbers and letters connected by +, – , ×, or ÷ ● examples: 5, r, 8t, x + 9, 2y – 5 variable ● a letter that represents an unknown number or amount A fish tank holds 3 algae eaters and some guppies. How many fish are there in total? The number of guppies is unknown. Number of guppies: g Number of algae eaters: 3 Total number of fish in the tank: g + 3 3. Write an expression. Tell what your variable describes. b) Shay has 5 packages of pencils. Each package has the same number of pencils. How many pencils are in all 5 packages? a) Simon has many shirts. He gives 2 shirts away. How many shirts does he still have? Variable: s = the number of Variable: Expression: s – Expression: Use a Coordinate Grid You can describe points on a coordinate grid using ordered pairs: (x, y). ● ● The x-coordinate tells you how many units to move left or right starting at the origin (0, 0). The y-coordinate tells you how many units to move up or down starting at the x-axis. y 2 A D B E C F 0 –2 –2 To plot the point E (3, 1), start at (0, 0). Move 3 units right and 1 unit up. x 2 G 4. Use the grid above. Write the coordinates of each point in the table of values. Point A B C D E x –1 3 y 0 1 F G Get Ready ● MHR 475 Name: _____________________________________________________ Date: ______________ Adventure Travel Have you ever wanted to climb a mountain? Via Ferrata is a steep mountain trail in Whistler, British Columbia. Paulette decided to climb the Via Ferrata trail. She recorded how long it took her to climb each 100-m section. Her times were 10 min, 9 min, 10 min, 11 min, 10 min, and 30 min. a) Complete the table with Paulette’s data. 100-m Section Time to Climb That Section 1 2 3 10 9 10 4 5 6 b) Describe the pattern of the time it took Paulette to climb sections 1 to 5. _________________________________________________________________________ c) Give 2 reasons why you think the time for section 6 was so different from the others. • _______________________________________________________________________ • _______________________________________________________________________ d) Using the table, draw a graph of Paulette’s total distance climbed compared to her total time. Total Time (min) 10 19 29 40 50 80 y 80 70 Total Time (min) Total Distance (m) 100 200 300 400 500 600 60 50 40 30 20 10 0 e) Describe the patterns you see on your graph. 100 200 300 400 500 600 x Total Distance (m) _________________________________________________________________________ _________________________________________________________________________ 476 MHR ● Chapter 9: Linear Relations Name: _____________________________________________________ Date: ______________ 9.1 Warm Up Ask yourself: “Do I add or subtract?” 1. For each chart, describe a pattern to go from the input to the output. Input 2 4 3 8 Output 5 7 6 11 Pattern: Input 6 3 7 10 Output 5 2 6 9 Input 1 3 5 6 Pattern: Pattern: Ask yourself: “Do I multiply or divide?” 2. For each chart, describe a pattern to go from the input to the output. Input 4 6 3 5 Output 8 12 6 10 Pattern: Input 6 12 3 15 Output 7 9 11 12 Output 2 4 1 5 Input 3 1 4 8 Pattern: Output 30 10 40 80 Pattern: 3. Write the coordinates of each point on the grid. A (3, 4) y 10 B( , ) F 9 8 C( , ) 7 C 6 D( , ) 5 B A 4 E( , ) 3 D 2 F( , ) 1 E 0 1 2 3 4 5 6 7 8 9 10 x 9.1 Warm Up ● MHR 477 Name: _____________________________________________________ Date: ______________ 9.1 Analysing Graphs of Linear Relations linear relation ● a pattern made by a set of points that lie in a straight line ● example: y linear = straight line 4 3 2 1 0 1 2 3 4 x table of values ● shows 2 sets of related numbers Working Example 1: Make a Table of Values From a Graph The graph shows that the total cost depends on the number of baseballs you buy. Total cost and number of baseballs are related to each other. C Cost of Baseballs 14 12 The variable b shows the number of baseballs. The variable C shows the total cost in dollars. Total Cost ($) 10 8 6 4 2 0 1 2 3 4 b Number of Baseballs a) Describe the patterns you see on the graph. Solution ● The graph shows data on the cost of baseballs. One ball costs $3, 2 balls cost ● The total cost increases by $ , 3 balls cost ,…. each time you buy a baseball. To move from 1 point to the next, go 1 unit horizontally (↔) and vertically (↕). ● 478 The points lie in a The graph shows a linear relation. MHR ● Chapter 9: Linear Relations units . Name: _____________________________________________________ Date: ______________ b) Make a table of values from the graph. Solution Read each point as an ordered pair to make a table of values. Complete each table. In a horizontal table of values, the top row shows the x-coordinates. The bottom row shows the y-coordinates. Number of 1 Baseballs (b) 2 Total Cost (C ) 6 3 3 In a vertical table of values, the first column shows the x-coordinates. The second column shows the y-coordinates. Number of Baseballs (b) 1 2 4 Total Cost (C ) 3 6 3 4 c) Write an expression that represents the cost of buying b baseballs. Solution Let b describe the number of baseballs. The expression 3b describes the cost of buying b baseballs. Number of Baseballs (b) 1 2 3 4 Total Cost (C ) 3 6 9 12 ×3 d) If the relationship in the graph continues, how much will it cost to buy 14 baseballs? Solution Use the expression 3b to find the cost of 14 baseballs. Cost of 14 baseballs = 3( ) ← Substitute 14 for b. 3(14) = 3 × 14 = The cost of 14 baseballs is $ . 9.1 Analysing Graphs of Linear Relations ● MHR 479 Name: _____________________________________________________ Date: ______________ a) Describe 2 patterns you see in the graph. Number of Squares ● s The points appear in a . ● To move from 1 point to the next: Squares in a Pattern 25 20 15 10 5 0 __________________________________________ 1 2 3 4 Figure Number b) The graph shows the number of squares in relation to the figure number. These figures match the graph. Draw Figure 4. Figure 1 Figure 2 Figure 3 Figure 4 c) Complete the table of values. Figure Number, f 1 2 3 4 Number of Squares, s d) Is this a linear relation? Circle YES or NO. e) Let f describe the figure number. Circle the expression that describes the number of squares in any figure: 5f or f + 5 f ) If the pattern continues, how many squares will there be in Figure 8? Sentence: ______________________________________________________________ 480 MHR ● Chapter 9: Linear Relations f Name: _____________________________________________________ Date: ______________ Working Example 2: Analyse Data on a Graph of a Linear Relation Nicole has a part-time job. The graph shows her pay related to the number of hours she works. P Nicole’s Rate of Pay 50 40 Pay ($) a) Describe the patterns you see in the graph. Solution ● 20 10 2 3 4 5 6 t Time (h) , the pay for 2 h is the pay for 3 h is 1 0 The graph shows data on the pay Nicole receives for each hour of work. The pay for 1 h is ● 30 , ,…. The points lie in a straight , so the graph is a linear relation. To move from 1 point to the next, move and 10 units vertically (↕). The Pay axis counts by 10s. b) Make a table of values. unit horizontally (↔) ● Solution Time (h) 1 Pay ($) 10 2 3 4 40 c) How much does Nicole make per hour? Solution Look at the table of values. In 1 h of work, Nicole makes $ . For each hour she works, Nicole makes $10 more, so she makes $10/hr. d) Is it possible to have points between the points shown on the graph? Explain why or why not. Solution Yes, it is possible to have points between the points on the graph. Nicole could work full hours and half hours. If she worked 2.5 h, she would be paid 2.5 × $10 = $ On the graph, this point would be (2.5, 25). . 9.1 Analysing Graphs of Linear Relations ● MHR 481 Name: _____________________________________________________ Chad is buying notebooks at Bob’s Bargain Store. The graph shows the cost of notebooks. Date: ______________ C Cost of Notebooks Cost ($) 8 a) Describe the 2 patterns you see in the graph. 6 4 2 ● The points appear to lie in a 0 2 3 4 5 n Number of Notebooks . ● 1 To move from 1 point to the next: _____________________________________________________________________ b) Complete the table of values. Number of Notebooks, n 1 2 3 4 Cost, C ($) c) Use the table to find the cost of 1 notebook. d) Write an expression to show how much n notebooks cost: e) How much would it cost to buy 52 notebooks? Expression → Substitute → Solve → Sentence: _____________________________________________________________ f) Is it possible to have points between the ones on the graph? Explain why or why not. _____________________________________________________________________ _____________________________________________________________________ 482 MHR ● Chapter 9: Linear Relations Name: _____________________________________________________ Date: ______________ 1. For each graph, is it possible to have points between the ones on the graph? Explain. C Graph A Cost ($) Cost ($) 6 4 12 _____________________________________ 8 4 2 0 Graph A: _____________________________ Graph B 16 8 1 2 3 4 d Distance (km) 0 Graph B: _____________________________ 1 2 3 4 n _____________________________________ Number of Cartons of Ice Cream 2. The graph shows how much higher you get each time you go up a step of a staircase. a) Describe the 2 patterns you see in the graph. ● The pattern lies in a Total Height of Stairs 80 60 40 20 0 . ● h 100 Total Height (cm) C 1 2 3 4 5 6 s Number of Steps To move from 1 point to the next: _______________________________________________________________________ b) Use the graph to complete the table of values. Number of Steps 1 2 3 4 Total Height (cm) c) Describe the pattern in the table of values. The total height starts at cm and increases by cm. d) Write an expression for the total height after climbing s stairs: e) If the relationship in the graph continues, what is the total height on step 10? 9.1 Analysing Graphs of Linear Relations ● MHR 483 Name: _____________________________________________________ Date: ______________ 3. Tessa and Vince go shopping at Bulk Bin. The graph shows the cost of banana chips. C 360 Cost of Banana Chips at Bulk Bin a) Does the graph show a linear relation? Explain why or why not. ______________________________________________ Cost (¢) 300 240 180 120 60 ______________________________________________ 0 100 200 300 400 500 600 q Quantity (g) b) Describe 2 patterns shown on this graph. ● The pattern of the points: ___________________________________________________ ● To move from 1 point to the next: ____________________________________________ c) Complete the table of values for this graph. Quantity (g) 0 100 200 300 400 Cost ($) d) Can the graph show the cost of 250 g of banana chips? Explain your answer. _________________________________________________________________________ 4. a) Complete the table of values for the ordered pairs on the graph. y 7 x 1 y 6 5 4 3 b) Describe the 2 patterns you see in the graph. ● The pattern of the points: ____________________________________________ ● To move from 1 point to the next: ____________________________________________ c) Extend your table of values so the x-column goes to 9. d) If this pattern continues, what is the value of y when x = 9? 484 MHR ● Chapter 9: Linear Relations 2 1 0 1 2 3 x Name: _____________________________________________________ Date: ______________ 5. The graph shows the rate of pay based on the number of hours worked. P Rate of Pay 80 Rate of pay means how much money you are paid for 1 h of work. 70 Pay ($) 60 50 40 30 20 10 1 0 2 3 4 5 6 t Time Worked (h) a) Make a table of values for the ordered pairs on the graph. Time Worked (h) Pay ($) 1 Look at the graph. b) What is the hourly rate of pay? c) If the time worked is 4.5 h, how much pay is earned? 6. The graph shows part of a linear relation that describes the cost of cake flower decorations. C Cost of Flowers Cost (¢) 120 90 60 30 0 1 2 3 4 f Number of Flowers Ask yourself, “Can I buy 2 flowers?” Is it reasonable to have points between the ones on the graph? Explain your answer. ____________________________________________________________________________ 9.1 Analysing Graphs of Linear Relations ● MHR 485 Name: _____________________________________________________ Date: ______________ Do you like adventures? There are many different adventure tours in Canada. a) You are going on a polar bear tour. The graph shows the cost of the trip. Describe 2 patterns on the graph. 2400 The overall pattern: ___________________________________________ ● Polar Bear Adventure 2800 Cost ($) ● C 3200 2000 1600 1200 To move from 1 point to the next: 800 400 ___________________________________________ 0 b) Complete the table of values for this graph. Number of Days 0 1 2 1 2 3 4 5 6 7 d Number of Days 3 4 5 6 7 Cost ($) c) What would the cost be for 8 days on the tour? d) The tour company offers a deluxe tour that includes food and clothing supplies. They charge $300 for signing up plus $400 for each day on the tour. Complete the table of values for the deluxe tour. Number of Days Cost ($) 0 1 2 3 4 5 6 7 300 + 400 e) Compare the data in the 2 tables of values. How are they alike? _______________________________________________________ ________________________________________________________________________ How are they different? ____________________________________________________ ________________________________________________________________________ 486 MHR ● Chapter 9: Linear Relations Name: _____________________________________________________ Date: ______________ 9.2 Warm Up 1. Graph each set of points. Input 2 4 3 8 Output 5 7 6 11 y 12 10 Output a) 8 6 4 2 0 2 4 6 8 x +, –, ×, or ÷ Input To go from the input to the output, the pattern is b) x 6 3 7 10 y 5 2 6 9 . y 10 8 6 4 2 0 2 4 6 8 10 x To go from x to y, the pattern is . 2. Solve. a) 12 ÷ 4 = b) 6 × 5 = c) 13 – 9 = d) 12 + 5 = 3. Fill in the boxes. a) 6 × = 12 b) 5 × = 15 c) 2 × = 18 d) 2 × –1=5 e) 3 × + 5 = 11 f) 5 × –4=6 Do multiplication first, and then add or subtract. 9.2 Warm Up ● MHR 487 Name: _____________________________________________________ Date: ______________ 9.2 Patterns in a Table of Values Working Example 1: Identify the Relationship in a Table of Values relationship • a pattern formed by 2 sets of numbers The pattern in this table of values describes a linear relation. A 0 1 2 3 4 a) Graph the ordered pairs. Solution The ordered pairs are (0, 0), (1, 3), ( ( , , ), and ( B , ), B 0 3 6 9 12 ). Plot the last 3 ordered pairs. 12 10 8 6 4 Consecutive numbers go in order from smallest to largest (e.g: 6, 7, 8, 9 …). 2 0 1 2 3 4 A b) What is the difference between consecutive A-values? Solution 3–2= 2–1= 1–0= Consecutive A-values have a difference of . c) What is the difference between consecutive B-values? Solution 9–6= 6–3= 3–0= Consecutive B-values have a difference of . d) Look at the graph. Describe how to move from (0, 0) to the next point. Solution Starting at (0, 0), move vertically (↕). 488 MHR ● Chapter 9: Linear Relations unit horizontally (↔) and units Name: _____________________________________________________ Date: ______________ e) Write an expression for B in terms of A. Solution Look at the table of values: To get from A to B, multiply by Look at the graph: When A increases by There are 3 ways to write B in terms of A: . , B increases by 3. Words Ordered Pair (x, y) Expression B is 3 times A (A, 3A) 3 × A or 3A This table of values represents a linear relation. a) Complete the table. Then, graph each ordered pair. x y 1 5 2 10 3 15 4 20 y Ordered Pair (x, y) 25 20 15 10 5 O 1 2 3 4 5 x × b) What is the difference in consecutive values of x and y? x y 4–3= 20 – 15 = 3–2= 15 – 10 = When x increases by y increases by , . c) Write an expression for y in terms of x. To get from x to y, multiply x by An expression for y in terms of x is y = . × x. 9.2 Patterns in a Table of Values ● MHR 489 Name: _____________________________________________________ Date: ______________ Working Example 2: Use a Table to Determine a Linear Relation Table 1 4 7 2 3 x y 6 11 8 15 m n 1 1 Table 2 2 4 3 7 4 8 a) Complete the chart to show the pattern in the values for each variable. Solution Table 1 Difference Between Consecutive First Variables Difference Between Consecutive Second Variables Table 2 2 7–3= 4–1= 11 – 7 = 7–4= 15 – 11 = The y-values have a difference 8–7= The n-values have differences of of 3, 3, and . . b) Graph each table of values. Which relation is linear? Solution Finish graphing the points. y Graph of Table 1 n 18 10 15 8 12 6 9 4 6 2 3 0 0 2 4 6 Graph of Table 2 1 2 3 The graph of Table linear relation. shows a 4 m 8 x c) Look at the graphs. Table 1 is a linear relation and Table 2 is not. Explain how you know. Solution Table 1: The difference in the y-coordinates is . The relation is linear. (the same or not the same) Table 2: The difference in the n-coordinates is . The relation is not linear. (the same or not the same) 490 MHR ● Chapter 9: Linear Relations Name: _____________________________________________________ Table 1: 1 1 x y 4 10 p q 7 19 0 0 Date: ______________ Table 2: 1 1 2 3 3 6 a) Complete the chart to find the difference between the consecutive values of each variable. Table 1 Difference Between Consecutive First Variables Table 2 4–1= – = 7–4= Consecutive x-values have a – = difference of . – = Consecutive p-values have a difference of Difference 10 – 1 = Between – = Consecutive Consecutive y-values have a Second Variables difference of . . 1–0= – = – The q-values differ by , = , . b) Is Table 1 linear? Explain. _______________________________________________________________________ Is Table 2 linear? Explain. _______________________________________________________________________ c) Check each answer by graphing. y Graph of Table 1 q 20 6 18 5 16 4 14 3 12 2 10 1 8 0 Graph of Table 2 1 2 3 4 5 6 p 6 4 2 0 2 4 6 8 x 9.2 Patterns in a Table of Values ● MHR 491 Name: _____________________________________________________ Date: ______________ Working Example 3: Use a Table of Values in Solving a Problem Sam is paid $7 for every hour of babysitting. a) The table of values shows how much she is paid for 1 h, 2 h, and 3 h of babysitting. Complete the table. Solution Number of Hours, n 1 2 3 Sam’s Pay, P 7 14 21 4 5 b) Is this a linear relation? Explain how you know. Solution Look at the table of values. Consecutive n-values have a difference of , or they increase by . Consecutive P-values have a difference of , or they increase by . This relation is linear because: • the number of hours (n) changes by the same amount • Sam’s pay (P) changes by the same amount c) Graph this relation. Solution Plot the numbers in the table of values on the grid. P 42 Sam’s Pay 35 28 21 14 7 0 492 1 2 3 4 5 Number of Hours 6 n MHR ● Chapter 9: Linear Relations Name: _____________________________________________________ Date: ______________ d) Write an expression to show Sam’s pay. Solution Look at the table of values. To get from n to P, multiply by Look at the graph. When n increases by . , P increases by . There are 3 ways to write Sam’s pay in terms of the number of hours she babysits: Words P is ________ times n Ordered Pair Expression n) n (n, e) How much will Sam be paid for 9 h of babysitting? Solution Sam’s pay = 7 × n =7× Substitute 9 for n. = Sam will be paid $ . Sky sells magazine subscriptions. She gets $20 for every 5 subscriptions she sells. a) Complete the table. Number of Subscriptions, n 0 5 10 Pay, P 0 20 40 15 20 25 b) Is this a linear relation? Explain how you know. • difference of consecutive n-values = The relation is • difference of consecutive P-values = . (linear or not linear) c) Let the number of subscriptions be n. An expression for Sam’s pay is . d) How much will Sky get paid if she sells 40 subscriptions? Sky’s pay = 4n 9.2 Patterns in a Table of Values ● MHR 493 Name: _____________________________________________________ Date: ______________ 1. Giselle and Tim are discussing the table of values. 3 1 m a 5 3 7 5 9 7 a) Who is correct? Circle GISELLE or TIM. b) How do you know? 2. x 1 2 3 4 a) The difference between consecutive x-values is y 5 8 11 14 (4 – 3 = , 3 – 2 =, , and 2 – 1 = b) The difference between consecutive y-values is (14 – 11 = . , 11 – 8 = , and ) . ) c) Does this table of values describe a linear relation? Circle YES or NO. Give 1 reason for your answer. _________________________________________________________________________ d) Graph the table of values. e) Look at the graph. Describe the movement from (1, 5) to the next point. y 14 Starting at (1, 5), move 12 10 unit horizontally and 8 6 units vertically. 4 2 0 494 1 2 3 4 x MHR ● Chapter 9: Linear Relations Name: _____________________________________________________ Date: ______________ 3. The table of values describes a linear relation. 0 0 x a 1 4 2 8 3 12 4 16 5 20 a) The difference between consecutive x-values is . b) The difference between consecutive a-values is . c) Graph the ordered pairs. d) Look at the graph. Describe in words how to move from (0, 0) to the next point. a 20 ___________________________________ 16 12 ___________________________________ 8 4 0 ___________________________________ 1 2 3 4 5 x e) Write a in terms of x. Words a is 4. x 2 3 4 5 times x Ordered Pair Expression x) x (x, y 7 10 13 16 a) What is the difference between consecutive x-values? b) What is the difference between consecutive y-values? Is the difference the same for consecutive values? Circle YES or NO. c) Is the relationship in the table of values a linear relation? Circle YES or NO. Give 1 reason for your answer. d) Check by graphing. y 16 _________________________________ 14 _________________________________ 10 12 8 _________________________________ _________________________________ 6 4 2 0 1 2 3 4 5 x 9.2 Patterns in a Table of Values ● MHR 495 Name: _____________________________________________________ Date: ______________ 5. Mara reads 90 words per minute. a) Complete the table of values. Number of Minutes, m 1 2 3 4 5 6 Number of Words, w b) Explain how you can find out if this is a linear relation. ________________________________________________________________________ ________________________________________________________________________ c) If the number of minutes is m, then the expression for the number of words is × m. d) How many words can Mara read in 15 min? Sentence: ___________________________________________________________________ 6. A community centre has a new banquet hall. The centre charges $5 per person to rent the hall. a) Complete the table of values. Number of People, p 1 Rental Cost, C ($) 5 20 40 60 80 100 b) If the number of people is p, then the expression for the rental cost is c) How much will it cost for 150 people? Sentence: __________________________________________________________________ 496 MHR ● Chapter 9: Linear Relations . Name: _____________________________________________________ Date: ______________ Plan a canoe trip. You can rent canoes at many national parks. One canoe costs $40 a day to rent. A park pass costs $36 for 1 week. a) Complete the table. The first row has been done for you. Number of Days, d Cost, C ($) b) Graph the ordered pairs in your table. Label the axes. y Ordered Pair 320 300 36 + (40 × 1) 1 280 260 (1, 76) = 36 + 40 240 = 76 220 200 36 + (40 × 2) 2 = 36 + ______ 180 (2, _____) 160 = ______ 140 120 100 3 80 60 4 40 20 0 5 1 2 3 4 5 6 7 8 x 6 7 c) Is this a linear relation? Circle YES or NO. Give 1 reason for your answer. ________________________________________________________________________ d) Write an expression for the cost, C, based on the number of days, d. C= + ×d 9.2 Math Link ● MHR 497 Name: _____________________________________________________ Date: ______________ 9.3 Warm Up 1. Complete the patterns. a) 3, 8, 13, , c) 10, 16, 22, , , b) 9, 7, 5, , d) 38, 35, 32, , 2. Whole numbers start at 0 and increase by 1 each time. a) List the first 10 even whole numbers. ________________________________ b) List the first 10 odd whole numbers. ________________________________ 3. Integers are … –3, –2, –1, 0, 1, 2, 3 , … Zero is not positive or negative. a) List the first 5 positive integers. ________________________________ b) List the first 5 negative integers. ________________________________ 4. Use substitution to evaluate. b) 8 + 2x, when x = 3 a) 3x – 4, when x = 5 3( = )–4 –4 Multiply. =8+ = = d) –7x – 1, when x = 0 c) –5x + 6, when x = 2 –5( = )+6 +6 = 5. Let n describe Nancy’s age. The expression n – 5 represents Jane’s age. If Nancy is 12 years old, how old is Jane? Jane is 498 8+2( Substitute. years old. MHR ● Chapter 9: Linear Relations ) Substitute. Multiply. Name: _____________________________________________________ Date: ______________ 9.3 Linear Relationships Working Example 1: Graph From a Linear Formula formula ● an equation that shows how 1 variable is related to another ● example: P = 2l + 2w shows how the perimeter of a rectangle is related to its length and width A car travels at 60 km/h. Use the formula d = 60t to express this relationship, where d is the distance travelled, in kilometres, and t is the time, in hours. a) Make a table of values. Time cannot be negative in this question. So, t cannot be a negative integer. Solution 0 Time, t 1 2 3 Distance, d To find d, substitute each value of t from the table into the formula d = 60t. d = 60t d = 60(0) d = 60t d = 60(1) d = 60t d = 60t d = 60( ) d = 60( ) d= d= d= d= b) Graph the ordered pairs in your table of values. Solution d Distance (km) 180 Speed of Car 150 120 90 60 30 O 1 2 3 4 t Time (h) c) Is it possible to have points between the ones on this graph? Solution 105 km is halfway between 90 km and 120 km. It is possible to have points between the ones on the graph. The distance at 2.5 h is km. The time at 105 km is h. 9.3 Linear Relationships ● MHR 499 Name: _____________________________________________________ Date: ______________ d) How far will the car travel in 3.5 h? Solution d = 60t d = 60 × ( ) Substitute 3.5 for t. d= The car will travel 210 km in 3.5 h. Paul rents a lawn mower for $8 per hour. The formula C = 8t describes the relationship between the rental cost and the time. C describes the rental cost and t describes time, in hours. a) Complete the table of values. t C What are possible values for t ? b) Graph your ordered pairs. To draw a graph: Label the x-axis “t” and the y-axis “C.” Write a description along each axis. Mark the intervals on the x- and y- axes. Give the graph a title. Plot the points. c) Find the rental cost for 12 h. 500 MHR ● Chapter 9: Linear Relations Name: _____________________________________________________ Date: ______________ Working Example 2: Graph From a Linear Equation Using Integers Integers are … –3, –2, –1, 0, 1, 2, 3, … . equation ● 2 expressions that have the same value and are joined with an equal sign ● examples: 2x + 3 = 10 and y = x – 5 y = –3x + 4 is a linear equation. a) Use the linear equation to make a table of values. Solution You do not know what x describes, so, use integer values for x. To find y, substitute each of the values for x into the equation y = –3x + 4. x –2 –1 0 y= 1 For x = 1: y = –3x + 4 y = –3(1) + 4 2 3 y= y= For x = 0: y = –3x + 4 y = –3(0) + 4 For x = –1: y = –3x + 4 y = –3(–1) + 4 For x = –2: y = –3x + 4 y = –3(–2) + 4 y=6+4 y = 10 y 10 +4 y= +4 +4 y= y= For x = 2: y = –3x + 4 y = –3(2) + 4 For x = 3: y = –3x + 4 y = –3(3) + 4 y= y= +4 +4 y= y= b) Graph the ordered pairs. c) Find the y-value in the ordered pair (11, y). Solution Solution The point (11, y) tells you that the x-value is 11. Substitute 11 into the formula y = –3x + 4. y 10 8 y = –3x + 4 6 4 y = –3( 2 –2 –1 0 –2 –4 –6 1 2 3 4 x y= )+4 +4 Substitute. Multiply. y= The value of the y-coordinate is The ordered pair is (11, . ). 9.3 Linear Relationships ● MHR 501 Name: _____________________________________________________ Date: ______________ Use y = 2x + 3 to answer the questions. a) Make a table of values using integers. x –2 –1 0 1 2 3 b) Graph the ordered pairs in your table of values. y y 10 9 8 7 6 5 4 3 2 1 Let x = –2 y = 2x + 3 y = 2(–2) + 3 y = –4 + 3 y = –1 Let x = –1 y = 2x + 3 Let x = 0 -5 -4 -3 -2 -1 O -1 2 3 4 5 x -2 -3 -4 y = 2( ___ ) + 3 -5 -6 -7 y = _____ + 3 -8 -9 y = ______ Let x = 1 1 Let x = 2 -10 Let x = 3 c) What are the coordinates for the point that lies on the y-axis? x = 0 for all points that lie on the y-axis. y = 2x + 3 y = 2( y= )+3 + y= The coordinates for the point that lies on the y-axis are (0, 502 MHR ● Chapter 9: Linear Relations ). Name: _____________________________________________________ Date: ______________ 1. A store sells hats for $10 each. Let h describe the number of hats. The formula S = 10h shows the amount of money collected in sales, S. a) Can negative integers be used for the values of h? Give 1 reason for your answer. _______________________________________________________________________ b) Can whole numbers be used for the values of h? Give 1 reason for your answer. _______________________________________________________________________ 2. a) Name the point that is 3 units to the right of the origin. ( , ) (0, 0) b) Name the point that is 4 units to the left of the origin. ( , ) c) What do you notice about these 2 points? ___________________________________ d) Is this true for all points that lie on the x-axis? Give 1 reason for your answer. _______________________________________________________________________ 3. Find the value of each equation. a) y = 5x – 3 when x = 6 y = 5( y= y= c) y = x – 8 when x = 5 b) y = 3x + 2 when x = –4 )–3 –3 ← Substitute → y = 3( ← Multiply → y= )+2 +2 y= d) y = –5x when x = –2 9.3 Linear Relationships ● MHR 503 Name: _____________________________________________________ 4. Use the formula C = 6t to describe a long-distance telephone plan, where C is the cost in cents and t is the time in minutes. a) Make a table of values. Use at least 4 whole number values for t. t 1 1 2 3 4 5 6 7 8 9 * 0 # Date: ______________ Only 6¢ per minute anytime for calls across Canada! CallCanada Let t = 1 C = 6t C = 6(1) C C= b) Graph the ordered pairs from your table of values. y c) If you round part minutes up to the next whole minute, is it possible to have points between the ones on your graph? Explain. ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ x 5. Complete each table of values. a) y = 3x + 2 x –2 0 2 4 y Let y = –2 y = 3(–2) + 2 y= y= b) y = –4x x –2 0 2 4 504 y MHR ● Chapter 9: Linear Relations +2 Let y = 0 Let y = 2 Let y = 4 Name: _____________________________________________________ Date: ______________ 6. An animal shelter pays you $5 for each dog you walk. Use the formula M = 5d to relate the money you make to the number of dogs you walk. M is the money you make and d is the number of dogs you walk. b) Graph the ordered pairs. a) Make a table of values. y d M x 7. For each equation, find the value of y in the ordered pair (2.5, y). a) y = 3x + 2 b) y = x – 5 y = 3(2.5) + 2 y= ← Substitute → +2 ← Multiply → y= 8. The graph shows Nigel’s monthly pay. e Nigel’s Monthly Earnings Monthly pay ($) 1600 A break in the y-axis means the length of the axis has been shortened. It can be shown as 1500 1400 1300 1200 0 1000 2000 3000 4000 5000 6000 7000 s Sales ($) a) If Nigel does not make any sales, what is his monthly pay? b) Nigel has sales of $4000 in 1 month. How much does he make? c) Nigel earns $1300 in 1 month. What are his sales? 9.3 Linear Relationships ● MHR 505 Name: _____________________________________________________ Date: ______________ 9. You are given part of a table of values for a linear relation. –3 x –2 –1 y 0 1 2 6 8 10 a) How could you find the missing y-coordinates? _________________________________________________________________________ b) Complete the table. 10. You can buy work gloves from a web site. Use the formula C = 5g + 2 to find the price. C is the cost in dollars and g is the number of pairs of gloves. a) Complete the table of values using whole numbers. g C b) Graph the ordered pairs. To draw a graph: Label each axis using g and C. Describe each axis. Mark the intervals on each axis. Give the graph a title. Plot the points. c) Is this a linear relation? Circle YES or NO. Give 1 reason for your answer. ___________________________________ ___________________________________ ___________________________________ d) Are other points possible between the ones on your graph? Circle YES or NO. Give 1 reason for your answer. _________________________________________________________________________ _________________________________________________________________________ 506 MHR ● Chapter 9: Linear Relations Name: _____________________________________________________ Date: ______________ 9 Chapter Review Key Words For #1 to #5, fill in the blanks. Unscramble the letters for each term to complete the sentence. 1. A pattern that creates points that lie in a straight line is called a relation. EILRAN 2. A chart showing the relationship between 2 sets of numbers is called a . LEBTA FO SUAVLE 3. In the expression 5g – 2, g is called a . ABEVIRAL 4. When 2 expressions are joined with an equal sign, you have an QONATUIE . 5. An equation that shows how 1 variable is related to another is called a . MALUROF 9.1 Analysing Graphs of Linear Relations, pages 478–486 6. Klaus works after school. The graph shows his rate of pay. P 45 a) Fill in the table of values. 1 2 y 36 Pay ($) x 0 Klaus’s Rate of Pay 27 18 9 0 1 2 3 4 t Time (h) 3 4 b) Does the graph represent a linear relation? Circle YES or NO. Give 1 reason for your answer. _________________________________________________________________________ c) Is it possible to have other points between the ones on the graph? Circle YES or NO. Give 1 reason for your answer. _________________________________________________________________________ Chapter Review ● MHR 507 Name: _____________________________________________________ 7. The graph shows the amount of money a grade 8 class made while doing a car wash fundraiser. I Money Collected ($) 1 2 3 Grade 8 Car Wash 50 a) Using the graph, fill in the table of values. Number of Cars Date: ______________ 4 40 30 20 10 Money Collected ($) 0 1 2 3 4 5 c Number of Cars b) Describe the 2 patterns you see in the graph. • The overall pattern: ________________________________________________________ • To move from 1 point to the next: _____________________________________________ c) Write an expression that describes the amount of money collected after washing c cars. d) If the students wash 15 cars, how much money will they collect? Amount collected = 10c 10c = 10 × c = 10 × = 9.2 Patterns in a Table of Values, pages 488–497 8. a) Is this a linear relation? A 0 1 2 3 4 5 B 0 7 14 21 28 35 The difference between consecutive A-values is (1 – 0 = ,2–1= . ,3–2= ) The difference between consecutive B-values is (7 – 0 = , 14 – 7 = . , 21 – 14 = ) The relation is , since the A-values change by the same amount and the B-values change by the same amount. b) Write B in terms of A. Words B is ________ times A 508 MHR ● Chapter 9: Linear Relations Ordered Pair (A, A) Expression A Name: _____________________________________________________ Date: ______________ 9. The table of values shows a relation. p 1 2 3 4 5 a) Is this a linear relation? Q 5 8 10 13 15 The difference between consecutive p-values is . The difference between consecutive Q-values is . The relation is , since ______________________________________________________. b) Graph the ordered pairs to check your answer. q 16 14 12 10 8 6 4 2 O 1 2 3 4 5 p 10. A recreation centre charges $5 per person to use the gym. a) Complete the table of values. Number of People 3 5 7 9 Gym Charge b) Without graphing, explain if this is a linear relation or not. __________________________________________________________________________ __________________________________________________________________________ c) Let n describe the number of people. Write an expression for the cost to use the gym. d) How much will it cost for 25 people to use the gym? Sentence: ____________________________________________________________________ Chapter Review ● MHR 509 Name: _____________________________________________________ Date: ______________ 9.3 Linear Relationships, pages 499–506 11. Craig travels at a constant speed of 15 km/h. The formula d = 15t describes the relationship. His speed does not change. a) What does each variable describe? d describes the t describes the b) Make a table of values. Use 5 consecutive whole number values for t. t 2 d Let t = 2 d = 15t d = 15(2) d= c) Graph your ordered pairs. To draw a graph: Label each of the axes using t and d. Describe each axis. Mark the intervals on both axes. Give the graph a title. Plot the points. d) Is it reasonable to have points between the ones on the graph? Circle YES or NO. Explain. __________________________________________________________________________ __________________________________________________________________________ e) How far would Craig travel in 8 h? Sentence: _________________________________________________________________ 510 MHR ● Chapter 9: Linear Relations Name: _____________________________________________________ Date: ______________ 12. y = 2x + 4 a) Use 5 integers for x to complete this table of values (use positive and negative integers). x Let y = −2 y = 2x + 4 y = 2(−2) + 4 y –2 y= +4 y= b) Graph the ordered pairs. y 10 9 8 7 6 5 4 3 2 1 -2 -1 O 1 2 3 4 5 6 7 8 9 10 x c) Find the value of y in the ordered pair (2.5, y). y = 2x + 4 y = 2( y= )+4 +4 Substitute. Add. y= d) Find the value of y in the ordered pair (–6, y). Chapter Review ● MHR 511 Name: _____________________________________________________ Date: ______________ 9 Practice Test For #1 to #5, choose the best answer. 1. You can describe 2x – 1 as a(n) . A constant C expression B equation D variable 2. The table shows the number of toothpicks in the base of a triangle in relation to its perimeter. Toothpicks in Base, b 1 2 3 Toothpicks in Perimeter, p 3 6 9 Which expression describes the number of toothpicks in the perimeter of any triangle in this pattern? A b+3 b C 3 B 3b D b–3 3. Which table of values describes this linear relation? y 8 6 4 2 0 512 1 2 3 4 x A x y 0 8 1 6 2 6 3 5 4 4 B x y 0 8 1 7 2 6 3 4 4 2 C x y 0 8 1 7 2 6 3 5 4 4 D x y 0 8 1 6 2 4 3 3 4 2 MHR ● Chapter 9: Linear Relations Name: _____________________________________________________ 4. Which table of values describes the linear equation y = 3x – 2? A x 1 2 3 y 1 4 8 B x 0 2 4 y 2 8 1 C x 2 3 4 y 4 7 10 D x 3 5 7 y 9 15 21 Date: ______________ Substitute values for x into the equation. 5. Which graph describes this statement: “A banquet room rents for $50 a night plus $2 per person”? The formula for the total cost is C = 50 + 2p, where p is the number of people. 10 20 30 40 n Number of People 10 20 30 40 n Number of People C 110 90 70 50 Cost ($) Cost ($) 0 D C 90 70 50 0 C 110 90 70 50 Cost ($) 90 70 50 0 C B C Cost ($) A 10 20 30 40 n Number of People 0 10 20 30 40 n Number of People For #6 and #7, complete the statements. 6. If the equation is y = 4x + 2, the value for y in (1, y) is 7. For this the graph, when the x-coordinate increases by 1, the y-coordinate . y 8 6 by . 4 2 0 1 2 3 4 x Practice Test ● MHR 513 Name: _____________________________________________________ Date: ______________ Short Answer 8. The graph shows the cost of a new drink called Zap. C a) What is the cost of 1 can of Zap? Cost of Zap 18 15 b) Describe 2 patterns in the graph. Cost ($) 12 9 • The pattern of the points: ____________________ 6 • To move from 1 point to the next: _____________ 3 0 1 2 3 4 5 6 n _____________________________________ Number of Cans c) Complete the table of values. Number of Cans, n 1 2 3 d) Describe 2 patterns in the table of values. The difference between consecutive values of Cost ($), C 3 6 9 n is . The difference between consecutive values of C is 4 . 5 9. The formula b = 4f represents the pattern in the table. The number of black dots is b and the figure number is f. Figure 1 Figure 2 Figure 3 a) Complete the table of values. Figure Number, f Number of Black Dots, b 1 2 3 4 5 514 MHR ● Chapter 9: Linear Relations b) Use the formula to find the number of black dots in Figure 10. Name: _____________________________________________________ Date: ______________ You decide to go on an adventure tour! Choose your adventure: • hang gliding • white-water rafting • any other adventure • hiking • dogsledding • canoeing • cycling a) Describe 3 different adventure tours you find on the Internet: 1. _____________________________________________________________________ 2. _____________________________________________________________________ 3. _____________________________________________________________________ b) Choose 1 of the adventures from part a): c) Using travel brochures, the Internet, or other resources, find or create data for a linear relation. d) Graph the ordered pairs in your table of values. Examples: • cost related to the number of hours • cost related to the rental charges • distance travelled over time • height climbed over time Record your findings in a table of values. e) Is it possible to have points between the ones on your graph? Explain why or why not. ________________________________________________________________________ Practice Test ● MHR 515 Name: _____________________________________________________ Date: ______________ Draw a line from the term in column A to the example in column B. Then, find each term in column A in the word search below. A B A. equation 1. P = 2l + 2w, where P is the perimeter, l is the length of a rectangle, and w is the width of the rectangle B. expression 2. C. formula D. linear relation 3. 0 20 Height (m) Temperature (°C) 15 30 45 y 15 12 E. table of values 9 6 F. variable 3 1 0 2 3 4 5 6 x 4. y = 4x 5. An unknown number, c 6. 6n – 2 E J O Y N M P N N D U A N N L I N E A R R E L A T B V J J W M K O Q B K I O N I K A N O I S S E R P X I C S R F O R M U L A F R M N O X O C P V B C I A G O H R M J P Q K J J I V D B E X T F U O V R B Y E D S I I W Z G E T K E U L A V F O E L B A T L S Z N Q O P E L X K S G C 516 T MHR ● Chapter 9: Linear Relations I 60 75 Name: _____________________________________________________ Date: ______________ Math Games −4 Friends and Relations −3 Play the Friend and Relations game with a partner. 3 4 2 1 −2 −1 0 • spinner BLM • paper clip • set of 20 Friends and Relations game cards • Friends and Relations chart for each student Rules: • Each player spins the spinner once to decide who will deal the cards. If there is a tie, spin again. • The dealer shuffles and deals 10 Friends and Relations game cards to each player. The other player takes the first turn. For each turn: • Flip over the top card in your stack. Copy the linear relation from the card into the Linear Relation column of your chart. • Spin the spinner and record the result in the x-column of your chart. • Calculate the y-value by substituting the x-value into the linear relation. Have your partner check your calculation. • Record the y-value in the y-column of your chart. • After each turn, add your y-values. Compare the totals. My y-value was 14 in my first turn and –8 in my second turn. My total score after 2 turns is 14 + (–8), which equals 6. How to win: • The player with the highest total score is the winner. • If there is a tie after 10 turns, shuffle the deck again and deal the cards. Take more turns until 1 player wins. Linear Relation x y 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. TOTAL of y-values Math Games ● MHR 517 Name: _____________________________________________________ Date: ______________ Challenge in Real Life Comparing Wages Five people work at Moy’s Food Mart. The table shows a weekly time sheet. • grid paper • calculator Total Hours Hourly Wage $12.50 Employee Mr. Moy Mon 8 Tues 8 Wed 8 Thurs 8 Fri 9 Sat — Sun — Ms. Wong — 8 8 7 — 9 9 $9.50 Maria 8 5 — — 6 9 5 $7.50 Tom 4 4 4 4 4 8 8 $7.50 Jacob — — 5 7 7 5 5 $7.50 Total Wage TOTAL 1. For each employee, find the “Total Hours” and the “Total Wage.” Total wage = total hours × hourly wage 2. Who made the most money? 3. Who worked the most hours? and 4. What is the total wages paid out for the week? _______________ 5. Complete the tables of values to find the rates of pay. Mr. Moy Hours Wage 0 5 10 15 20 Ms. Wong Hours Wage 0 5 10 15 20 Maria Hours Wage 0 5 10 15 20 Mr. Moy’s hourly rate = $12.50 Wage for 0 h = 0 × 12.50 = Ms. Wong’s hourly rate Maria’s hourly rate = = Wage for 5 h = 5 × 12.50 = 6. Graph each table of values on a grid. Use a different colour for each set of points. 518 MHR ● Chapter 9: Linear Relations Answers 3. a) Yes, the graph is linear because the points lie on a straight line. Get Ready, pages 474–475 b) Answers may vary. Example: The points lie in a straight line; move 100 units right and 60 units up. 1. a) letters of the alphabet; begins with b; skips 2 letters b) integers; begins with 9; decreases by 5 2. a) 1 4 Figure Number Number of Squares 2 7 3 10 c) 1 4 Figure Number Number of Cubes 2 6 3 8 x 1 2 3 4 5 6 7 8 9 y 2 4 6 8 10 12 14 16 18 b) Answers may vary. Example: The points lie in a straight line; move 1 unit right and 2 units up d) 18 x –1 0 –1 3 3 3 2 y 2 1 0 2 1 0 –1 5. a) 1 2 3 4 5 6 10 9 10 11 10 30 b) Answers may vary. Example: It took Paulette approximately the same time for the first 5 sections, creating a consistent pattern. c) Answers may vary. Example: The terrain could have been more difficult, or Paulette could have taken a break. Paulette’s Climb t Total Time (min) 100 80 60 40 20 0 e) Answers may vary. Example: The first 5 dots appear to lie along a straight line, but the last dot does not. 200 400 600 d Total Distance (m) b) $15 c) $67.50 Math Link a) Answers may vary. Example: The points lie in a straight line; move 1 unit right and 400 units up. b) d) 2. multiply by 2; divide by 3; multiply by 10 Number of Days 0 1 2 3 4 5 6 7 Cost ($) 0 400 800 1200 1600 2000 2400 2800 Number of Days 0 1 2 3 4 5 6 7 Cost ($) 300 700 1100 1500 1900 2300 2700 3100 e) Answers may vary. Example: They are similar because the rate of increase is $400.00 per day. They are different because the starting value for the first table is (0, 0) and the starting value for the second table is (0, 300). 3. B (1, 5), C (0, 7), D (7, 2), E (6, 0), F (8, 9) 9.1 Analysing Graphs of Linear Relations, pages 478–486 Working Example 1: Show You Know 1 5 2 10 9.2 Warm Up, page 487 Figure 4 3 15 1. a) add 3 4 20 10 8 6 4 2 0 Working Example 2: Show You Know 1 2 2 4 3 6 4 8 c) $2.00 d) 2n e) $104.00 f) No; you cannot buy part of a book. 2 4 6 8 10 x x 1. Answers may vary. Example: Graph A: Yes, you can travel part of a kilometre. Graph B: No, you cannot buy part of a carton of ice cream. Practise 2. a) Answers may vary. Example: straight line; move 1 unit right and 20 units up 3 60 3. a) 2 b) 3 c) 9 d) 3 e) 2 f) 2 9.2 Patterns in a Table of Values, pages 488–497 Working Example 1: Show You Know a) Communicate the Ideas 2 40 2 4 6 8 Input 2. a) 3 b) 30 c) 4 d) 17 a) straight line; move 1 unit right and 2 units up 1 b) Number of Steps 20 Total Height (cm) c) 20 cm; 20 cm d) 20 s e) 200 cm y 12 10 8 6 4 2 0 d) YES e) 5f f) 40 squares Number of Notebooks, n Cost, C ($) b) subtract 1 y Output a) straight line; move 1 unit right and 5 units up b) b) Pay ($) 15 30 45 60 75 6. Yes, you can buy 2 flowers. 1. add 3; subtract 1; add 6 Figure Number, f Number of Squares, s Time Worked (h) 1 2 3 4 5 c) $3200 9.1 Warm Up, page 477 c) 400 240 G Math Link d) 300 180 4. a) and c) 3. a) shirts; s – 2 b) Let p stand for the number of pencils; 5p 4. Point A B C D E F 100-m Section Time to Climb That Section 200 120 Apply Start with 4 cubes and increase by 2. a) 100 60 d) Answers may vary. Example: Yes, you can have a point between the ones on the graph because you can buy between 100 and 200 grams. Start with 4 squares and increase by 3. b) 0 0 Quantity (g) Cost ($) 4 80 x 1 2 3 4 y 5 10 15 20 Ordered Pair (x, y) 1, 5 2, 10 3, 15 4, 20 y 35 30 25 20 15 10 5 0 1 2 3 4 x b) 1, 5 c) y = 5x Answers ● MHR 519 Apply Working Example 2: Show You Know a) Difference Between Consecutive First Variables Difference Between Consecutive Second Variables Table 1 4−1=3 7−4=3 Consecutive x-values have a difference of 3. Table 2 1−0=1 2−1=1 3−2=1 Consecutive p-values have a difference of 1. 10 − 1 = 9 19 − 10 = 9 Consecutive y-values have a difference of 9. 1−0=1 3−1=2 6−3=3 The q-values differ by 1, 2, 3. b) Yes, Table 1 is linear because the difference in consecutive values are the same; No, Table 2 is not linear because the difference in consecutive values of q is not the same. of Table 1 c) Graph y 5. a) Number of Minutes, m Number of Words, w 6. a) 1 2 3 4 5 6p 5 10 15 20 25 0 20 40 60 80 100 b) linear c) 4n d) $160 1. a) TIM b) Answers may vary. Example: The relation is linear because all consecutive values of m differ by 2 and values of a differ by 2. Practise 2. a) 1 b) 3 c) YES. The relation is linear because all consecutive values of x differ by 1, and all consecutive values of y differ by 3. e) 1 unit, 3 d 14 12 10 8 6 4 2 450 540 Number of People, p Rental Cost, C ($) 1 20 40 60 80 100 5 100 200 300 400 500 Cost, C ($) 76 116 156 196 236 276 316 c) YES. This relationship is linear because there is a common difference between the consecutive values for both variables. d) C = 36 + 40 × d 320 300 280 260 240 220 200 180 160 140 120 100 80 60 40 20 0 1 2 3 4 5 6 7 Number of Days Ordered Pair (# of Days, Cost) (1, 76) (2, 116) (3, 156) (4, 196) (5, 236) (6, 276) (7, 316) n 9.3 Warm Up, page 498 3. a) 1, 2, 3, 4, 5 b) –1, –2, –3, –4, –5 4. a) 11 b) 14 c) 4 d) –1 1 2 3 4 5a 5. 7 d) move 1 unit right, and 4 units up e) four, 4, 4 9.3 Linear Relationships, pages 499–506 Working Example 1: Show You Know a) 1 2 3 4 5x 4. a) 1 b) 3; YES c) YES. The relation is linear because all consecutive values of x differ by 1, and all consecutive values of y differ by 3. d t 1 2 3 4 C 8 16 24 32 Cost of Lawnmover Rental b) C Rental Cost ($) A 20 16 12 8 4 0 520 360 1. a) 18, 23, 28 b) 3, 1 c) 28, 34 d) 29, 26 3. a) 1 b) 4 c) 40 32 24 16 8 0 16 14 12 10 8 6 4 2 0 270 2. a) 0, 2, 4, 6, 8, 10, 12, 14, 16, 18 b) 1, 3, 5, 7, 9, 11, 13, 15, 17, 19 0 d) 180 Cost of Canoe Trip Communicate the Ideas d) 90 C Total Cost ($) 0 6 Number of Days, d 1 2 3 4 5 6 7 b) Number of Subscriptions, n Pay, P 5 a) Working Example 3: Show You Know a) 4 b) 5p c) $750 2 4 6 8x 0 3 Math Link Graph of Table 2 0 2 b) Answers may vary. Example: You can find out if this is a linear relationship by checking the consecutive values for m and w. c) 90 d) 1350 q 6 5 4 3 2 1 20 18 16 14 12 10 8 6 4 2 1 1 2 3 4 5n MHR ● Chapter 9: Linear Relations 1 2 3 4 5 t Rental Time (h) c) $96 Working Example 2: Show You Know a) x –2 –1 0 1 2 3 Chapter Review, pages 507–511 b) y –1 1 3 5 7 9 c) (0, 3) y 10 9 8 7 6 5 4 3 2 1 1. linear 2. table of values 3. variable 4. equation 5. formula 6. a) b) YES. The points on the graph are in a straight line. c) YES. Klaus could work part of his shift. y 0 9 18 27 36 1 10 7. a) Number of Cars Money Collected ($) 1 2 3 4 5x -2 0 -2 x 0 1 2 3 4 2 20 3 30 4 40 b) The points lie in a straight line. Move 1 unit right, and 10 units up. c) 10c d) $150 Communicate the Ideas 8. a) linear b) seven; 7; 7 1. a) No, because you cannot sell a negative number of hats. b) Yes, because you can sell any number of hats, beginning at 0 and going up. 9. a) not linear; the changes in consecutive values of Q are not the same. b) 2. a) (3, 0) b) (–4, 0) c) The second coordinate is 0. d) Yes, because they lie in a straight line. Practise 3. a) y = 27 b) y = –10 c) y = –3 d) y = 10 C 6 12 18 24 b) Cost of Long Distance 30 Phone Plan C 0 24 18 12 6 10. a) c) No, because any part minutes are rounded up to the next nearest minute. 6. a) x –2 0 2 4 y –4 2 8 14 1 5 d M 2 10 b) 3 15 x –2 0 2 4 4 20 M Money Earned 35 30 25 20 15 10 5 for Dog Walking 0 1 2 3 4 5 6d Number of Dogs g 1 2 3 4 C 7 12 17 22 –2 2 b) –1 4 0 6 1 8 2 10 Cost of Work Total Cost ($) 10. a) –3 0 9 45 c) d 30 45 60 75 90 Distance Travelled by a Cyclist d 90 75 60 45 30 15 2 4 Time (h) 6 t d) YES. Craig can travel for a length of time that is not a whole number. e) 120 km 8. a) $1200 b) $1400 c) $2000 x y t 2 3 4 5 6 0 Apply 9. a) Use a pattern. b) 7 35 c) 5n d) $125 b) b) 7. a) y = 9.5 b) y = –2.5 5 25 11. a) distance travelled in kilometres; time in hours y 8 0 –8 –16 Money ($) 5. a) 3 15 Number of People Gym Charge b) Yes, this is a linear equation because the consecutive values of both variables have a common difference. 1 2 3 4 t Time (min) 0 1 2 3 4 5p Distance (km) t 1 2 3 4 Total Cost (¢) 4. a) q 16 14 12 10 8 6 4 2 12. a) Answers may vary. Example: x –2 –1 0 1 2 y 0 2 4 6 8 b) Answers may vary. Example: y 10 8 6 4 –2 0 2 4 6 8 10 x c) y = 9 d) y = –8 C Gloves 20 10 0 2 4 g Pairs of Gloves c) YES. The points are in a straight line. d) NO. The values of g must be whole numbers because they represent the number of pairs of gloves. Answers ● MHR 521 Practice Test, pages 512–514 2. Mr. Moy 1. C 2. B 3. C 4. C 5. D 3. Mr. Moy; Ms. Wong 6. 6 4. $1637.00 7. decreases; 1 5. 8. a) $3.00 b) Answers may vary. Example: The points appear to lie in a straight line; The values for both variables have a constant difference; move 1 unit right and 3 units up. c) Number of Cans, n 1 2 3 4 5 9. a) Cost ($), C 3 6 9 12 15 Number of Black Dots, b 4 8 12 16 20 Figure Number 1 2 3 4 5 Mr. Moy Hours 0 5 10 15 20 d) 1; 3 Maria Hours 0 5 10 15 20 b) 40 6. Wrap It Up!, page 515 Wages ($) a)–e) Answers will vary. Key Word Builder, page 516 1. C 2. E 3. D 4. A 5. F 6. B Challenge in Real Life, page 518 1. Employee Mr. Moy Ms. Wong Maria Tom Jacob TOTAL 522 Total Hours 41 41 33 36 29 180 Hourly Wage $12.50 $9.50 $7.50 $7.50 $7.50 Wage $0 $62.50 $125.00 $187.50 $250.00 Total Wage $512.50 $389.50 $247.50 $270.00 $217.50 $1637.00 MHR ● Chapter 9: Linear Relations w 260 240 220 200 180 160 140 120 100 80 60 40 20 0 Wage $0 $37.50 $75.00 $112.50 $150.00 Wages at Mr. Moy’s Food Mart 5 10 15 Hours Worked 20 h Mr. Moy Ms. Wong Maria Ms. Wong Hours Wage 0 $0 5 $47.50 10 $95.00 15 $142.50 20 $190.00
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