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Practice Lesson 17 Solve Problems U

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©Curriculum Associates, LLC Copying is not permitted. 175Lesson 17 Solve Problems Using Systems of Equations

Name:

Lesson 17

Solve Problems Using Systems of Equations

Prerequisite: Use Substitution to Solve Systems of Equations

Study the example problem showing how to use substitution to solve a system of equations. Then solve problems 1–7.

1 Explain why you substitute x 1 4 for y in the fi rst equation of the system in the example

2 Once you know the value of one variable in a system of equations, how can you fi nd the value of the second variable?

3 Look at the system of equations at the right Which variable would you fi nd the value of fi rst? Explain your reasoning and solve for that variable

Example

Use substitution to solve this system of equations

y 1 3x 5 24 y 5 x 1 4

The second equation tells you Now you can find the value of y that y 5 x 1 4, so you can substitute You can substitute 22 for x intox 1 4 for y in the first equation and either equation and solve for y solve for x Try using the second equation

y 1 3x 5 24 y 5 x 1 4 (x 1 4) 1 3x 5 24 4x 1 4 5 24 y 5 22 1 4 4x 5 28 y 5 2 x 5 22

The solution is (22, 2)

4y 1 x 5 12x 5 2y

175175

B

B

You substitute x 1 4 for y to make an equation with just x in it.

You substitute that value into one of the equations and solve for the second variable.

Possible answer: y; I know that x 5 2y, so I can substitute 2y for x in the first equation and

solve for y. 4y 1 2y 5 12; 6y 5 12; y 5 2

M

©Curriculum Associates, LLC Copying is not permitted.176 Lesson 17 Solve Problems Using Systems of Equations

Solve.

4 Use substitution to solve the system of equations

2y 2 x 5 29 y 5 2x 2 3

Show your work.

Solution:

Use the system of equations at the right for problems 5–6.

5 Graph the system of equations What ordered pair appears to be the solution?

6 Solve the system of equations algebraically to check your solution to problem 5

Show your work.

Solution:

7 Tom’s work to solve a system of equations is shown Do you agree with Tom’s statement about the solution? Explain Describe the graph of the system of equations

y 5 2x 2 4y 5 2x 1 2

System Using Substitution

y 5 22x 1 1 2x 1 (22x 1 1) 5 3

2x 1 y 5 3 1 5 3

The system has no solution

x

y

O2324 22 21

22

24

23

211 2 3 4

1

2

3

4

176

M

M

M

C

2y 2 x 5 –9 y 5 2x 2 3

2(2x 2 3) 2 x 5 29 y 5 2(21) 2 3

4x 2 6 2 x 5 29 y 5 22 2 3

3x 2 6 5 29 y 5 25

3x 5 23

x 5 21

(21, 25)

2x 2 4 5 2x 1 2 y 5 2x 2 4

3x 2 4 5 2 y 5 2(2) 2 4

3x 5 6 y 5 4 2 4

x 5 2 y 5 0

(2, 0)

(2, 0)

Yes; The statement 1 5 3 is not true, so there is no

solution. The graph of the system are parallel lines

that do not intersect.

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Unit 3

Practice Lesson 17 Solve Problems U

sing Systems of Equations

©Curriculum Associates, LLC Copying is not permitted. 177Lesson 17 Solve Problems Using Systems of Equations

Name:

Solve Real-World Problems

Study the example problem showing how to use a system of equations to solve a real-world problem. Then solve problems 1–7.

Lesson 17

1 Explain what the equation 110d 5 100d 1 40 represents in the context of the example problem

2 Suppose Oceanview Hotel changes their fee to $45 and Beachside Hotel changes their daily rate to $115 Write new equations for the total costs for the two resorts

3 Solve the system of equations formed by the equations you wrote in problem 2 After how many days would the total costs at the two resorts be the same?

Example

Oceanview Hotel charges $100 per day plus a one-time fee of $40 Beachside Hotel charges $110 per day After how many days will the costs at the two hotels be equal?

Start by writing a system of equations to model the problem Let c be the cost and d be the number of days

Total cost for Oceanview: c 5 100d 1 40Total cost for Beachside: c 5 110d

Use substitution to solve the system The second equation tells you that c 5 110d, so you can substitute 110d for c

c 5 100d 1 40

110d 5 100d 1 40 Substitute 110d for c

10d 5 40

d 5 4

The costs at the hotels will be the same after 4 days

177

B

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M

It represents when the costs at the two hotels will be equal.

The equation for Oceanview would be c 5 100d 1 45, and the equation for Beachside

would be c 5 115d.

The total costs would be the same after 3 days.

©Curriculum Associates, LLC Copying is not permitted.178 Lesson 17 Solve Problems Using Systems of Equations

Solve.

4 Roberto got $30 for his birthday He decides to save that amount and add $5 to his savings each week Jack starts saving the same day as Roberto and puts $8 in his savings each week After how many weeks will the boys have the same amount in savings?

Show your work.

Solution:

Use this situation for problems 5–6.

Julia earns $6 an hour babysitting and earns $5 an hour walking dogs She earned $43 after working a total of 8 hours at her two jobs

5 Complete the system of equations below to represent the situation Let b 5 the number of hours that Julia babysits and d 5 the number of hours she walks dogs

1 5 8 1 5 43

6 Solve the system of equations from problem 5 to fi nd the number of hours Julia worked at each job

7 Consider the situation at the right Write a question and a system of equations for the situation Then answer your question by solving the system of equations

Trisha and Yoshi are at the start of a trail Trisha walks 500 feet before Yoshi starts Trisha walks 350 feet per minute, and Yoshi walks 430 feet per minute

178

Possible answer:Let s 5 total savings and w 5 number of weeks.Roberto’s savings: s 5 5w 1 30Jack’s savings: s 5 8w8w 5 5w 1 30 3w 5 30 w 5 10

They will have the same amount in savings after 10 weeks.

Julia worked 3 hours babysitting and 5 hours walking dogs.

Possible answer: How long will it take Yoshi to

catch up with Trisha?

Let d 5 distance and t 5 time.

Trisha: d 5 350t 1 500 and Yoshi: d 5 430t

It will take 6.25 minutes for Yoshi to catch up with Trisha.

b d 6b 5d

M

M

C

B

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Unit 3 Exp

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Practice Lesson 17 Solve Problems U

sing Systems of Equations

©Curriculum Associates, LLC Copying is not permitted. 179Lesson 17 Solve Problems Using Systems of Equations

Name: Lesson 17

1 The sum of two numbers is 27 One number is 3 more than the other number Write and solve a system of equations to fi nd the two numbers

Show your work.

Solution:

3 Use the system of equations you wrote in problem 2 to fi nd how many dimes and quarters Roberta has

A 10 dimes and 15 quarters

B 10 dimes and 35 quarters

C 15 dimes and 10 quarters

D 15 dimes and 35 quarters

Dennis chose A as the correct answer How did he get that answer?

2 Roberta has $4 00 in dimes and quarters She has 5 more dimes than quarters Write a system of equations that you could use to fi nd how many dimes and quarters she has

Write one equation for the sum. What will the other equation be?

Choose a variable for the number of dimes and a variable for the number of quarters.

Check your solution in the equation that shows the total amount of money.

Solve Problems Using Systems of Equations

Solve the problems.

179

Possible answer: Let x 5 one number and y 5 the other number.x 1 y 5 27x 5 y 1 3

(y 1 3) 1 y 5 27 x 1 12 5 27 2y 1 3 5 27 x 5 15 2y 5 24 y 5 12

The numbers are 12 and 15.

Possible answer: Let d 5 the number of dimes

and q 5 the number of quarters.

d 5 q 1 5 and 10d 1 25q 5 400

Dennis switched the number of dimes and the number of quarters.

M

B

M

©Curriculum Associates, LLC Copying is not permitted.180 Lesson 17 Solve Problems Using Systems of Equations

Solve.

4 Line a passes through the points (23, 22) and (0, 4) Line b passes through the points (22, 23) and (0, 1) Tell whether each statement is True or False

a. Lines a and b intersect u True u False

b. Lines a and b have different slopes u True u False

c. Lines a and b have different y-intercepts u True u False

d. Lines a and b are parallel u True u False

5 The Parks and Recreation Department in your town off ers a season pass for $150

• With the season pass you pay $5 per session to use the town’s tennis courts

• Without the season pass you pay $15 per session to use the tennis courts

Part A

Write a system of equations to represent the situation

Part B

Graph your system of equations How many times do you need to use the tennis courts for the season pass to save you money? Explain

Solution:

What does the point where the lines intersect represent?

Remember that the y-intercept is the y-coordinate when the x-coordinate is 0.

Co

st ($

)

Sessions

10 20 305 15 25Os

c

50

25

100

75

125

150

175

200

225

250

275

180

3

3

33

Possible answer: Let s 5 the number of sessions and c 5 the total cost.

with season pass: c 5 5s + 150; without season pass: c 5 15s

16 times; Playing tennis 15 times will cost

you the same, $225, with or without the season pass.

Playing 16 times will cost you $230 with the pass

and $240 without.

M

C

Without season pass

Withseason pass