Algebra 1 Topice 10: Systems of linear equations and ...

Student: Class: Date:

Algebra 1 Topice 10: Systems of linear equations and inequalities page 1

Student Activity Sheet 1; use with Overview

1. Consider the equation y = 3x + 4.
a. What can you know about the graph of the equation?

b. How does a point, (x,y), end up on the graph of this equation?

2. Consider the equation 2x + 2y = 24.
a. What do you know about the graph of this equation?

b. How does a point, (x,y) end up on the graph of this equation?

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Algebra 1 Topice 10: Systems of linear equations and inequalities page 2

Student Activity Sheet 1; use with Overview

3. Show why the point (2,10) is a solution to the system of equations.
y = 3x + 4

2x + 2y = 24

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Algebra 1 Topice 10: Systems of linear equations and inequalities page 3

Student Activity Sheet 2; use with Exploring “Solving systems of equations in slope-intercept
form”

1. Desmond has determined the following information about his two options.

Option 1, repairing his current gas-powered mower: The cost of repair will be $160, after
which the cost to operate the repaired mower is estimated to be $45 per week.

Option 2, buying a new energy-efficient electric-powered mower: The cost of the new
mower will be $400, after which the cost to operate the new mower is estimated to be
$15 per week.

Create equations to model the options. Each equation should model the cost of that
option, y, in dollars, as a function of time, x, in weeks.

400 15 45 160

2. Desmond’s best option will be that option with the smallest total cost over the rest of the
mowing season.
a. Which option is best if there is little time left in the mowing season?

b. Which option is probably best if the mowing season has just begun?

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Algebra 1 Topice 10: Systems of linear equations and inequalities page 4

Student Activity Sheet 2; use with Exploring “Solving systems of equations in slope-intercept
form”

c. It would be useful to know how many weeks it would take for the two options to
result in the same total cost. How can Desmond figure that out?

3. What is the solution to the following system?
y = 45x + 160
y = 15x + 400

4. What does the solution you found in question 3 tell you about Desmond’s decision with
respect to his two options?

5. The two ad options are

$10 per week plus 12.5¢ per click
$6.25 per week plus 20¢ per click

Set up an equation to model the total cost for each option if Desmond runs the ad for one
week. Then use your graphing calculator to help you determine how many clicks it would
take for the two options to result in the same total cost.

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Content copyright 2014 Charles A. Dana With space for student work
Center, The University of Texas at Austin

Student: Class: Date:

Algebra 1 Topice 10: Systems of linear equations and inequalities page 5

Student Activity Sheet 2; use with Exploring “Solving systems of equations in slope-intercept
form”

6. REINFORCE Andrew has challenged his cousin, Kyle, to a race. Since Andrew is older,

Kyle insists on a head start. Andrew agrees and gives Kyle a 20-meter head start. Andrew
runs at a constant rate of 8.5 meters per second, and Kyle runs at a constant rate of 6
meters per second.

a. Set up a system of two linear equations whose solution will represent at what
distance and at what time Andrew will overtake Kyle in the race.

b. Solve the system using both a table and a graph. Interpret the solution.
c. Show how both the table and the graph can be used to determine how long it takes

before Andrew is 10 meters behind Kyle.

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Content copyright 2014 Charles A. Dana With space for student work
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Student: Class: Date:

Algebra 1 Topice 10: Systems of linear equations and inequalities page 6

Student Activity Sheet 3; use with Exploring “Solving systems of equations in standard form”

1. Desmond has lost his record sheet for the second week of his work. The only thing he
remembers is that he received $240 for mowing 12 lawns that week.

In the neighborhood where Desmond and his customers live, the houses are built on lots
that come in two sizes: standard-sized interior lots and larger corner lots. Desmond charges
$15 per standard-sized lot and $30 per large corner lot for his mowing services.
If x represents the number of standard-sized lots mowed that week and y represents the
number of large corner lots mowed that week, complete the puzzle to create a system of
two linear equations to model this situation.

240 15x + 30y 12 x+y

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Content copyright 2014 Charles A. Dana With space for student work
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Student: Class: Date:

Algebra 1 Topice 10: Systems of linear equations and inequalities page 7

Student Activity Sheet 3; use with Exploring “Solving systems of equations in standard form”

2. You found a system of equations to represent Desmond’s missing records situation.
a. How is this system different from the systems of equations you solved in the last
Exploring?

b. Will this difference prevent you from using a table and/or a graph to solve the
system?

c. How do you know whether to use slope-intercept form or standard form for the
equations in system of linear equations?

3. Sketch the graph of each equation.
x + y = 12

15x + 30y = 240

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Content copyright 2014 Charles A. Dana With space for student work
Center, The University of Texas at Austin

Student: Class: Date:

Algebra 1 Topice 10: Systems of linear equations and inequalities page 8

Student Activity Sheet 3; use with Exploring “Solving systems of equations in standard form”

4. As you saw in both the table and graph, the solution to the system is the point (8,4).
What does this solution tell you about Desmond’s missing records problem regarding how
many lots of each size were mowed during the week?

5. Suppose this time, Desmond charges $20 for the standard-sized interior lots and $25 for
the larger corner lots. He received a total of $300 for mowing 14 lots during the past
week. Using x to represent the number of interior lots mowed and y to represent the
number of corner lots mowed, set up a system of two linear equations to model this
situation. Then use your graphing calculator to help you determine how many lots of
each type Desmond mowed during the week.

6. REINFORCE The sum of two numbers is 186. The difference between the same numbers is

32. Create a system of equations to model the situation, and then solve the system using
a graphical strategy.

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Student: Class: Date:

Algebra 1 Topice 10: Systems of linear equations and inequalities page 9

Student Activity Sheet 4; use with Exploring “Systems of inequalities”

1. Desmond needs to schedule the lawn mowing jobs each week so that he and Shelly can
do all of the mowing and edging that is required. What are some issues that Desmond
needs to consider?

2. Based on their previous experience, Desmond and Shelly come up with the following time
estimates:

Standard-sized interior lot: 1 hour to mow and a half hour to edge
Larger corner lot: 2 hours to mow and 45 minutes to edge

Desmond can spend at most 30 hours a week mowing lawns. Shelly can only spend at
most 12 hours per week edging.

How can Desmond model this information as a system of two linear inequalities?

Standard-sized yards Large yards Constraints

Number of yards

Number of mowing
hours per yard

Number of edging
hours per yard

System of inequalities:

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Student: Class: Date:

Algebra 1 Topice 10: Systems of linear equations and inequalities page 10

Student Activity Sheet 4; use with Exploring “Systems of inequalities”

3. Determine whether the following combinations of sizes of lawns represent feasible
numbers of lawns to mow and edge in any one week, given the time constraints
represented by the system of inequalities. Explain your conclusions in terms of the
amount of time Desmond and Shelly will work in each case.
a. 10 standard-size interior lawns and 8 large corner lawns

b. 6 standard-size interior lawns and 15 large corner lawns

c. 6 standard-size interior lawns and 12 large corner lawns

d. 18 standard-size lawns and 6 large corner lawns

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Student: Class: Date:

Algebra 1 Topice 10: Systems of linear equations and inequalities page 11

Student Activity Sheet 4; use with Exploring “Systems of inequalities”

4. Sketch the graph of x + 2y ≤ 30. 18
16
5. Sketch the graph of 1 x + 3 y ≤ 12 . 14
2 4 12
10

8
6
4
2

10 20 30

18
16
14
12
10

8
6
4
2

10 20 30

6. Sketch the graph of the system: 18
16
x + 2y ≤ 30 14
12
1 x + 3 y ≤ 12 10
2 4
8
6 20 30
4
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Student: Class: Date:

Algebra 1 Topice 10: Systems of linear equations and inequalities page 12

Student Activity Sheet 4; use with Exploring “Systems of inequalities”

7. Desmond makes two important observations.
a. “Because neither variable can be negative, we really have a system of four
inequalities, not just two inequalities.”
Write these two new inequalities, along with the original two inequalities, to show
the complete system of four inequalities to which Desmond is referring.

b. Desmond continues, “Also notice that the point (10,8) is below both of the lines,

x + 2y ≤ 30 and 1 x + 3 y ≤ 12 . That agrees with the observation we made earlier
2 4

about the mowing schedule that point represents!”

Can you explain what Desmond means by this?

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Content copyright 2014 Charles A. Dana With space for student work
Center, The University of Texas at Austin

Student: Class: Date:

Algebra 1 Topice 10: Systems of linear equations and inequalities page 13

Student Activity Sheet 4; use with Exploring “Systems of inequalities”

8. Complete the statements to explain the other points representing combinations of sizes
of lawns. Use the answer choices provided.

Shelly above both below Desmond neither on

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Algebra 1 Topice 10: Systems of linear equations and inequalities page 14

Student Activity Sheet 4; use with Exploring “Systems of inequalities”

9. Desmond and Shelly decide to allocate more time per week to their respective tasks.
Desmond increases his mowing time to a maximum of 36 hours per week. Shelly increases
her edging time to a maximum of 15 hours per week. Write the system of inequalities
that represents the new constraints. Graph the system of inequalities, shading the
solution set. Are the mowing schedules represented by the points (6,15) and (18,6) now
feasible?

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Student: Class: Date:

Algebra 1 Topice 10: Systems of linear equations and inequalities page 15

Student Activity Sheet 4; use with Exploring “Systems of inequalities”

10.REINFORCE The souvenir shop at the ballpark sells signed baseballs for $2 each and

signed miniature bats for $6 each. Jay can spend at most $12 to buy no more than 4
items. Create a system of inequalities to model this situation, and then graph the
solution set. Which solution results in Jay spending all his money to buy 4 items?

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Content copyright 2014 Charles A. Dana With space for student work
Center, The University of Texas at Austin