Chapter 1 Diagnostic Test - Nelson 1 Diagnostic Test STUDENT BOOK PAGES 4–7 1. Complete the table...

Chapter 1 Diagnostic Test STUDENT BOOK PAGES 4–7

1. Complete the table of values for the linear relation 3x + 2y = 18. x –2 4 0 10

y 12 6 –3 0 2. Graph the relation 5y – 4x = 60 by determining its

x- and y-intercepts.

3. a) Using a graphing calculator, enter the relation y = –3x + 7 into the equation editor as Y1. Press GRAPH to view the graph of the relation.

b) Determine the slope and y-intercept of the relation 12x + 4y = 16. Predict how the graph of this relation would compare with the graph of the relation in part a).

c) Write the relation in part b) in the form y = mx + b, enter it as Y2, and press GRAPH. Explain what you see.

4. a) To graph the equation 3x – 6y = 9, would you determine the x- and y-intercepts or would you determine the slope and y-intercept? Explain your choice.

b) You suspect that the equations 4y – 3x + 12 = 0 and 134

=−yx have identical graphs.

Describe a strategy you could use to decide whether your suspicion is correct.

5. Expand and simplify as necessary. a) 12x – 10 – 7x – 5 b) 4(–2x + 7) c) (2x – 3) – (5x – 4) d) 3(x + 7) – 5(6 – 2x)

6. Solve. a) x – 5 = –3 b) 26 = –2x – 8 c) 3x – 10 = –2x d) 3x + 5 = 5x + 17

7. Consider the equations y = m1x + 4 and y = m2x – 2. For what values of m1 and m2 will the graphs of these two equations be identical? For what values will the graphs be parallel? Explain your answers.

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Chapter 1 Diagnostic Test | 1

Chapter 1 Diagnostic Test Answers 1. x –2 4 2 0 8 6 10

y 12 3 6 9 –3 0 –6

2.

x-intercept: –15; y-intercept: 12

3. a)

b) slope: –3; y-intercept: 4; same slope,

different y-intercept (i.e., parallel) c) y = –3x + 4

The two lines are parallel because they have the same slope but different y-intercepts.

4. a) Answers may vary, e.g., I prefer x- and y-intercepts because the equation will not need to be rearranged.

b) Answers may vary, e.g., I could determine the x- and y-intercepts of both lines, and then compare them to see if they are equal.

5. a) 5x – 15 c) –3x + 1 b) –8x + 28 d) 13x – 9

6. a) x = 2 c) x = 2 b) x = –17 d) x = –6

7. The graphs will never be identical because they have different y-intercepts. They will be parallel only when m1 = m2

because the slopes are the same for parallel lines.

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If students have difficulty with the questions on the Diagnostic Test, it may be necessary to review the following topics: • expanding and simplifying algebraic expressions • graphing a linear relation, given the x- and y-intercepts • graphing a linear equation, given the y-intercept and the slope • solving linear equations

2 | Principles of Mathematics 10: Chapter 1 Diagnostic Test Answers

Lesson 1.1 Extra Practice STUDENT BOOK PAGES 8–14

1. The ordered pair (c, c) satisfies the relation 3x – y = 8. What is the value of c?

2. Define variables for each situation, and write an equation to represent the situation. a) Alexia earns $7.00/h gardening for her aunt

and $8.50/h babysitting. Last month, she earned $98.50.

b) Dieter has a long-distance plan that costs $4.95/month, plus 4¢/min for long-distance calls within Canada and the U.S., and 5¢/min for calls to Europe. Last month, Dieter’s total spending on long-distance calls was $11.95.

3. Graph each equation for question 2.

4. Justin buys almonds at 2.25¢ per gram and dried apricots at 1.5¢ per gram to make a snack mix. He spends $6.00 in total. a) Write an equation to represent this situation,

using x for the amount of almonds, in grams, and y for the amount of dried apricots, in grams.

b) Use a graphing calculator to graph your equation. Remember to use appropriate window settings.

c) Set up and display a table of values for your equation to help you determine the amount of dried apricots Justin bought if he bought 250 g of almonds.

5. One year ago, Renata invested some of her income in two investment accounts: a guaranteed-income fund that pays a return of 4% per year and a high-growth account that, for the year just past, paid out 6%. Renata’s return on her investment is $360. Use two different strategies to represent Renata’s possible investments in each account.

6. Marta works for a company that makes floral arrangements for special events. She is negotiating a new plan for her travelling expenses and suggests 15¢/km as a fair rate. The company suggests an alternative plan of 12¢/km, with an additional flat payout of $12.50/month. In a typical month, Marta drives 700 km on company business. Which plan is better for Marta?

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Lesson 1.1 Extra Practice | 3

Lesson 1.1 Extra Practice Answers 1. c = 4

2. a) Let x represent the number of hours spent gardening, and let y represent the number of hours spent babysitting. 7.00x + 8.50y = 98.50

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b) Let x represent the number of minutes spent on calls within Canada or the U.S., and let y represent the number of minutes spent on calls to Europe. 0.04x + 0.05y + 4.95 = 11.95

3. a)

b)

4. a) 2.25x + 1.50y = 600 b)

c)

5. Answers may vary, e.g., Table of values:

x 0 1500 3000 4500 6000 7500 9000

y 6000 5000 4000 3000 2000 1000 0

Graph:

Equation: 0.04x + 0.06y = 360 Graphing calculator:

6. The plan that Marta suggested is better.

4 | Principles of Mathematics 10: Lesson 1.1 Extra Practice Answers


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1. DriveEasy, a car share-rental company, rents vehicles online on a sliding scale, as shown in the graph. a) How much does it cost to

rent a car for 1 h 45 min? b) For how long can you rent

a car if you have $18.00?

2. a) Write an equation for the linear relation in question 1.

b) Use your equation to calculate the answers for question 1.

c) Compare your original answers for question 1 with your answers for part b). Identify one advantage of each strategy.

3. This graph shows the relationship between the cost of a safari tour and the number of tourists.

a) Write an equation for the relation. b) Use your equation to determine the cost

of a safari for six people. c) How many tourists can go on a safari for

$1250? How many can go for 1850? d) Is it possible for a safari to cost $1600?

Explain.

4. Benoit drove at 85 km/h from Barrie to Sudbury. At this speed, his truck uses gas at a rate of 7.5 L/100 km. Benoit left Barrie with 40 L of gas in the tank, and his low fuel warning light came on when 4 L was left. Estimate how long after leaving Barrie the warning light came on.

5. Nita is comparing two job offers. Roxie’s Boutique is offering $3000/month plus 25% commission on sales, and Cherie Womenswear is offering $2400/month plus 40% commission on sales. Compare the two offers. Which job should Nita take? Explain your decision.

6. A train leaves Montréal for Toronto at 9:30 a.m., travelling at 120 km/h. At the same time, a train bound for Montréal leaves Toronto, travelling at 105 km/h. The distance from Montréal to Toronto by train is 520 km. a) Describe two strategies you could use to

estimate when the trains will pass each other. b) Use one of your strategies to estimate when

the trains will pass each other.


Lesson 1.2 Extra Practice Answers 1. a) about $8.60 b) about 4 h 20 min

2. a) y = 3.5x + 2.5 b) $8.63, 4 h 26 min

c) Answers may vary, e.g., using the graph to estimate is quicker, but using the equation is more accurate.

3. a) y = 800 + 150x

c) 3 tourists, 7 tourists b) $1700

d) no, because the corresponding x-value is not a whole number

4. Answers may vary, e.g., about 5 h 40 min

5. Roxie’s pays more than Cherie Womenswear for up to $4000/month in sales. If Nita thinks that she can sell more than $4000, she should choose Cherie Womenswear’s offer; otherwise she should choose Roxie’s offer.

6. a) Answers may vary, e.g., drawing a graph of the distances of the two trains from either Montréal or Toronto; creating a table of values of these distances and determining when these values are equal; writing expressions to represent the distances travelled by the trains, setting the sum of the two expressions for distance equal to 520, and solving for time.

b) Answers may vary, e.g., about 11:50 a.m.

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1. Which ordered pair is a solution to the system of equations x + 2y = 6 and y = 8 – 3x? a) (–2, 2) b) (0, 8) c) (0.5, 6.5) d) (2, 2)

2. Which system of equations matches the graph shown?

a) 3x + y = 2 and 2x – 3y = 6 b) 3x – y = 2 and 2x + 3y = 12 c) 2x – 3y = 12 and y = 3x – 2 d) y = 3x – 2 and 2x + 3y = 6

3. a) Graph the system 3x – y = 5 and x + 2y = 4 by hand.

b) Solve the system using your graph. c) Verify your solution by substituting into the

given equations.

4. Use graphing technology to graph and solve the system 2x – 5y = 3 and x + y = 12.

5. A passenger airplane takes 4 h 15 min for a journey of 3600 km. The airplane travels at a cruising speed of 900 km/h. The mean speed is 600 km/h during takeoff and landing. Use a graphing strategy to estimate the amount of time that the airplane travels at cruising speed.

6. Anya works as a senior sales rep for a computer store. She earns $4500/month plus 5% commission on her monthly sales, but she is considering an offer of $3750/month plus 10% commission from another store. a) Which option is better if Anya’s monthly

sales average $12 000? Which option is better for sales of $20 000?

b) Write equations for Anya’s two options, and graph your equations.

c) How much in monthly sales would Anya have to make for both options to have the same value?

7. Five years ago, a high-school cafeteria charged $5.85 for three pieces of fruit and a chicken salad. Today, each piece of fruit costs 12% more, while a chicken salad costs 15% more. The new cost of three pieces of fruit and a chicken salad is $6.66. Determine the new prices of a piece of fruit and a chicken salad.

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Lesson 1.3 Extra Practice Answers 1. d)

2. b)

3. a)

b) (2, 1) c) 3(2) – 1 = 5, 2 + 2(1) = 4

4.

5. 3 h 30 min

6. a) Anya’s current job; the other offer b) y = 4500 + 0.05x, y = 3750 + 0.10x

c) $15 000

7. 84¢, $4.14

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Chapter 1 Mid-Chapter Review Extra Practice STUDENT BOOK PAGES 30–32

1. Define variables x and y for each situation, and write an equation to represent the situation. a) Indra earns $20/h at her day job and $12/h at

her evening job. Last month, she earned $3600.

b) Laurent keeps a change jar for snack machines. The jar contains $15.75 in loonies and quarters.

c) Rebecca goes on a road trip, travelling at 100 km/h on six-lane highways and 80 km on other highways. She travels a total distance of 480 km.

2. Graph each equation for question 1.

3. Waterworld Rentals rents windsurfing boards for $32/day and regular surfboards for $20/day. Last Tuesday, Waterworld charged $960 for rentals. Choose two strategies to represent the possible combinations of windsurfing boards and regular surfboards.

4. This graph shows the scale of fares charged by Speedy Taxi Company.

a) What is the minimum fare for a Speedy Taxi trip?

b) What is the fare for a 12 km trip? c) How much extra does each kilometre cost? d) Write an equation to represent Speedy Taxi

fares. Use your equation to determine the fare for a 29 km trip.

5. A 1300 L water tank empties at the rate of 4 L/min. At 3:15 p.m., 170 L of the water is left in the tank. Estimate when the tank was last filled.

6. a) Graph the linear system 3x + 4y = 12 and y = 2x – 3 by hand.

b) Estimate the solution to this system. c) Check your answer for part b) using a

graphing calculator.

7. Elena runs an ice cream store. She sells a 1 L tub of vanilla ice cream for $2.50 and a 1 L tub of mocha ice cream for $3.30. She wants to create a mix called Creamy Coffee Swirl to sell for $3.00 per 1 L tub. How much of each flavour must Elena use for one tub?

8. The equations y = 4, 2x + 3y = 8, and 2x – y = 8 form the sides of a triangle.

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a) Graph the triangle, and determine the coordinates of the vertices.

b) Calculate the area of the triangle.

Chapter 1 Mid-Chapter Review Extra Practice | 9

Chapter 1 Mid-Chapter Review Extra Practice Answers 1. a) Let x represent hours worked at the day

job, let y represent hours worked at the evening job; 20x + 12y = 3600

b) Let x represent number of loonies, let y represent number of quarters; x + 0.25y = 15.75

c) Let x represent time driven on six-lane highways, let y represent time driven on other highways; 100x + 80y = 480

2. a)

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b)

c)

3. Answers may vary, e.g., Let x represent Graph: the number of windsurfing boards. Let y represent the number of regular surfboards.

Equation: 32x + 20y = 960

Table of values: x 0 5 10 15 20 25 30

y 48 40 32 24 16 8 0

Graphing calculator:

4. a) $4.50 b) $7.50 c) 25¢ d) y = 0.25x + 4.50, $11.75

5. about 10:30 a.m.

6. a)

b) about (2.2, 1.4) c)

7. 0.375 L of vanilla and 0.625 L of mocha

8. a) b) 16 square units

(–2, 4), (4, 0), (6, 4)

10 | Principles of Mathematics 10: Chapter 1 Mid-Chapter Review Extra Practice Answers


1. Isolate the indicated variable in each equation. a) 3x + y = 7, y c) 4x + 3y = 12, x b) y – 3x + 2 = 0, x d) 20x – 4y = 10, y

2. Solve the linear system x + y = 5 and 3x – 2y = 25.

3. A hat maker at a fair sells two kinds of novelty hats. The banana-split hat sells for $3.50, and the chocolate-sundae hat sells for $4.25. At the end of the day, the hat maker has sold 76 hats and taken in $290 in revenue. How many chocolate-sundae hats were sold?

4. The hat maker in question 3 has to pay $130/day to rent the stall at the fair, and the materials to make each hat cost $2.25. Determine how many hats per day the hat maker must sell to break even if a) only banana-split hats are sold b) only chocolate-sundae hats are sold

5. The difference between two adjacent angles in a parallelogram is 42°. Determine the measures of all four angles in the parallelogram.

6. Without graphing, determine the intersection point of the line 2x + y = 8 and the line passing through (0, 6) and (9, 0).

7. A change jar contains $7.55 in nickels, dimes, and quarters. There are eight more nickels and dimes than quarters, and 50 coins altogether. How many of each coin are in the jar?

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Lesson 1.4 Extra Practice Answers 1. a) y = 7 – 3x

b) x = 31 y +

32

c) x = 3 – 43 y

d) y = 5x – 2.5

2. (7, –2)

3. 32

4. a) 104 b) 65

5. 111°, 69°, 111°, 69°

6. (1.5, 5)

7. 12 nickels, 17 dimes, 21 quarters

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1. a) Add and subtract the equations in the linear system 5x – 3y = 6 and x + 2y = –4.

b) By graphing, verify that the two new equations have the same solution as the original linear system.

2. a) Multiply 3x – y = 5 by –2, and multiply

2x + 4y = 7 by 21 .

b) Make a prediction about the graphs of the two new equations, compared with the graphs of the original two equations.

c) Suggest and apply a strategy to check your prediction without graphing.

3. a) Multiply x + 3y = 5 by 2, and multiply 3x + 2y = 15 by 3.

b) Create another linear system by adding and subtracting your equations for part a).

c) By graphing, verify that your linear system for part b) is equivalent to the original linear system for part a).

4. The linear system 2x – 3y = 2 and 3y – x = 8 is equivalent to the linear system ax + 13y = –22 and 3x + by = –6. Determine the values of a and b.

5. A student committee sells 104 tickets for a concert. Student tickets cost $9, and non-student tickets cost $12.50. The total revenue from ticket sales is $1135.50. a) Write two equations for this situation: one

equation describing the number of tickets sold, and the other equation describing the revenue.

b) Multiply your equation for the number of tickets by 9. Then subtract this new equation from your revenue equation. How are your two new equations related to your equations for part a)?

c) Determine the number of student tickets that were sold. Explain your strategy.

6. a) Use a substitution strategy to solve the system y + 5x + 32 = 0 and 3y – 4x = 37.

b) Show that multiplying the first equation by 3 and then adding and subtracting the equations forms an equivalent system.

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Lesson 1.5 Extra Practice Answers 1. a) adding: 6x – y = 2;

subtracting: 4x – 5y = 10 b)

2. a) –6x + 2y = –10, x + 2y = 3.5 b) The graph of –6x + 2y = –10 will be the

same as the graph of 3x – y = 5; the graph of x + 2y = 3.5 will be the same as the graph of 2x + 4y = 7.

c) Answers may vary, e.g., check intercepts: –6x + 2y = –10 and 3x – y = 5 both have

intercepts ⎟⎠⎞

⎜⎝⎛ 0,

35 and (0, –5); x + 2y = 3.5

and 2x + 4y = 7 both have intercepts (3.5, 0) and (0, 1.75).

3. a) 2x + 6y = 10, 9x + 6y = 45 b) adding: 11x + 12y = 55;

subtracting: –7x = –35 c)

4. a = –10, b = –6

5. a) x + y = 104, 9.00x + 12.50y = 1135.50 b) 9x + 9y = 936, 3.50y = 199.50; equivalent

to equations for part a) c) 47; answers may vary, e.g., I solved the

equation 3.50y = 199.50 to determine that y = 57, substituted this value into the equation x + y = 104, and solved for x.

6. a) (–7, 3) b) 6y + 11x = –59, 19x = –133; the solution is

again (–7, 3), so the systems are equivalent.

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1. For an Open House at a high school, a student-run stall is selling two types of juice drinks: Power Juice and Juice Cooler. The table shows the amounts of pure juice and water in each type.

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Type of Drink Amount of Juice (mL)

Amount of Water (mL)

Power Juice 350 150

Juice Cooler 250 750

If the students have 5.5 L of juice and 7.5 L of water, how much of each type of drink can they make?

2. During her morning commute, Rebecca averaged 30 km/h in heavy city traffic and 90 km/h once she got onto the highway. She travelled 35 km in 30 min. How far did Rebecca drive while on the highway?

3. To eliminate x from each linear system, by what numbers would you multiply equations ① and ②? a) 2x – 3y = –1 ① c) 5x + y = 12 ①

4x + y = 9 ② –2x – 3y = 0 ② b) 7x – 5y = 8 ① d) 2x + 2y = 5 ①

y + 3x = 9 ② 3x – 2y = –3 ②

4. A linear system consists of the equations 3x – 2y + 10 = 0 and 5x + 4y = –13. a) Solve the system by eliminating x. b) Solve the system by eliminating y.

5. You need to solve the linear system 3x – 5y = 41 and 2y – 3x = –29. a) Explain which variable you would choose to

eliminate. b) Solve the system by eliminating the variable

you chose.

6. A local charity decides to invest $20 000 in two funds. After one year, the returns on the funds are 4% and 6%. If the total return on the charity’s investment is $1040, how much did the charity invest in the fund returning 4%?

7. Kara thinks of two numbers. She performs the following steps: Step 1: She doubles the first number and subtracts three times the second number. The result is 19. Step 2: She switches the numbers and repeats step 1. The result is –41. Use an elimination strategy to discover Kara’s two numbers.


Lesson 1.6 Extra Practice Answers 1. 5 L of Power Juice, 8 L of Juice Cooler

2. 30 km

3. a) 2 and 1 c) 2 and 5 b) 3 and 7 d) 3 and 2

4. a), b) (–3, 0.5)

5. a) I would eliminate x because I don’t need to multiply either equation by a number. I can just add the equations.

b) (7, –4)

6. $8000

7. 17, 5

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Chapter 1 Review Extra Practice STUDENT BOOK PAGES 60–63

1. Steve has a budget of $25 each month to spend on cell-phone calls and text messages. He pays 15¢/min for airtime and 25¢ per text message. a) Use a table to show Steve’s possible

combinations of calls and text messages in one month.

b) Draw a graph to represent Steve’s possible combinations.

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2. Carly buys a bulk lot of stuffed animals and resells them at the school fair. The graph shows Carly’s net profit or loss.

a) Estimate Carly’s profit or loss if she sells

37 stuffed animals. b) Use the graph to write an equation that

represents Carly’s profit or loss. c) Determine Carly’s exact profit or loss if she

sells 37 stuffed animals.

3. Which system of equations matches the graph shown?

a) 3x + y = 1 and 2x – 5y = 6 b) x + 3y = 1 and x – 2y = 3 c) 3x + y = 1 and 5y – 2x = 6 d) x + 3y = 1 and 2x – 5y = 6

4. a) Isolate y in both equations in the system 4y + 7x = 19 and 5x – 2y = 13.

b) Use graphing technology to solve the system, expressing your answers to the nearest hundredth.

5. A transversal crosses two parallel lines, creating angles that measure a and b as shown. Determine a and b if tripling the angle measure a creates the same measure as doubling the angle measure b.

6. Which linear system is not equivalent to the system y – 2x = 13 and x + 5y = 10? a) x – y = –8 and 3x + 2y = –9 b) 4x – 7y = 41 and y = 3 c) x = –5 and 3x + 5y = 0 d) x – y = –8 and x + y = –2

7. Jamal wants to make a 300 g serving of kiwi and blueberries that contains 75 mg of vitamin C. He knows that 1 g of kiwi contains 0.7 mg of vitamin C, and each gram of blueberries contains 0.1 mg of vitamin C. Use any appropriate strategy to determine how much of each type of fruit Jamal should include.

8. a) Solve the system 3x + 2y = –4 and 5x – 2y = 12 using a substitution strategy.

b) Use an elimination strategy to solve the same linear system.

c) Which strategy is more efficient for solving this system? Explain in terms of both strategies.

9. a) Given the equation 5y – 3x = 15, write another equation to create a linear system with each number of solutions.

i) none ii) one iii) infinitely many b) Verify your answers for part a) graphically.

Chapter 1 Review Extra Practice | 17

Chapter 1 Review Extra Practice Answers 1. a) b)

Calls Text Messages

Number of Minutes

Cost ($)

Number of Messages

Cost ($)

Total Cost ($)

0 0 100 25 25

20 3 88 22 25

40 6 76 19 25

60 9 64 16 25

80 12 52 13 25

100 15 40 10 25

120 18 28 7 25

140 21 16 4 25

160 24 4 1 25

2. a) loss of about $64 or $65

b) y = 4.75x – 240 c) loss of $64.25

3. a)

4. a) y = x47

419

− , y = 2

1325

−x

b)

(2.65, 0.12)

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5. a = 72°, b = 108°

6. b)

7. 75 g kiwi, 225 g blueberries

8. a), b) (1, –3.5) c) Elimination is more efficient because the two

equations can simply be added together. For substitution, isolating either variable involves dividing both sides of one of the equations by a number.

9. Answers may vary, e.g., a) i) 5y – 3x = 10 iii) 10y – 6x = 30 ii) y – 3x = 9 b)

18 | Principles of Mathematics 10: Chapter 1 Review Extra Practice Answers

Chapter 1 Diagnostic Test - Nelson 1 Diagnostic Test STUDENT BOOK PAGES 4–7 1. Complete the table...

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Transcript of Chapter 1 Diagnostic Test - Nelson 1 Diagnostic Test STUDENT BOOK PAGES 4–7 1. Complete the table...