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Solving Systems of Linear Equations: Linear Combinations Answers

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To eliminate the y-terms and solve for x in the fewest steps, by which constants should the equations be multiplied by before adding the equations together?First Equation: 5x − 4y = 28Second equation: 3x - 9y = 30

A
The first equation should be multiplied by 3 and the second equation by 5.
B
The first equation should be multiplied by 3 and the second equation by −5.
C
The first equation should be multiplied by 9 and the second equation by 4.
D
The first equation should be multiplied by 9 and the second equation by −4.
2
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The linear combination method is applied to a system of equations as shown.4(.25x + .5y = 3.75) → x + 2y = 15 (4x – 8y = 12) → x – 2y = 3 2x = 18

Question illustration
A
(1,2)
B
(3,9)
C
(5,5)
D
(9,3)
3

Which statement describes the graph of the system of equations below?1.5x + 0.2y = 2.681.6x + 0.3y = 2.98

A
The lines are parallel.
B
The lines overlap at all points.
C
The lines intersect at (1.6,1.4).
D
The lines intersect at (3.1,0.5).
4

Melinda and Paula shovel driveways and sidewalks in the winter as a way to earn extra money. Together they shoveled 450 square feet of sidewalk in 30 minutes. Then Melinda shoveled for 20 minutes while Paula shoveled for 25 minutes to complete 345 square feet of driveway.30x + 30y = 45020x + 25y = 345

A
3 square feet per minute
B
6 square feet per minute
C
9 square feet per minute
D
15 square feet per minute
5

To eliminate the x terms and solve for y in the fewest steps, by which constants should the equations be multiplied by before adding the equations together? First equation: Second equation:

Question illustration
A
The first equation should be multiplied by 3 and the second equation by -5.
B
The first equation should be multiplied by 3 and the second equation by 5.
C
The first equation should be multiplied by 7 and the second equation by -6.
D
The first equation should be multiplied by 7 and the second equation by 6.
6

How many solutions are there to the system of equations?

Question illustration
A
no solutions
B
one solution
C
two solutions
D
an infinite number of solutions
7

Karina uses the system of equations below to compare the monthly utility costs in July and December for electricity, x, and natural gas, y. 750x + 17y = 141.61300x + 30y = 75.90

A
The cost of natural gas is $0.17 per unit.
B
The cost of natural gas is $0.20 per unit.
C
The cost of natural gas is $0.72 per unit.
D
The cost of natural gas is $0.83 per unit.
8

A personal trainer designs exercise plans based on a combination of strength-training and aerobic exercise. A beginner plan has 15 minutes per session of strength-training and 20 minutes per session of aerobic exercise for a total of 90 minutes of exercise in a week. An advanced plan has 20 minutes per session of strength-training and 30 minutes of aerobic exercise for a total of 130 minutes of exercise in a week.

A
Each plan utilizes a combination of 2 strength-training sessions and 2 aerobic exercise sessions per week.
B
Each plan utilizes a combination of 2 strength-training sessions and 3 aerobic exercise sessions per week.
C
Each plan utilizes a combination of 3 strength-training sessions and 2 aerobic exercise sessions per week.
D
Each plan utilizes a combination of 3 strength-training sessions and 3 aerobic exercise sessions per week.
9

The system of equations shown is solved using the linear combination methodWhat does 0 = 0 mean regarding the solution to the system?

Question illustration
A
There are no solutions to the system because the equations represent parallel lines.
B
There are no solutions to the system because the equations represent the same line.
C
There are infinitely many solutions to the system because the equations represent parallel lines.
D
There are infinitely many solutions to the system because the equations represent the same line.
10

The system of equations is solved using the linear combination method.What does 0 = −12 mean regarding the solution to the system?

Question illustration
A
There are no solutions to the system because the equations represent parallel lines.
B
There are no solutions to the system because the equations represent the same line.
C
There are infinitely many solutions to the system because the equations represent parallel lines.
D
There are infinitely many solutions to the system because the equations represent the same line.
11

To eliminate the x terms and solve for y in the fewest steps, by which constants should the equations be multiplied by before adding the equations together? First equation: Second equation:

Question illustration
A
The first equation should be multiplied by 3 and the second equation by -5.
B
The first equation should be multiplied by 3 and the second equation by 5.
C
The first equation should be multiplied by 7 and the second equation by -6.
D
The first equation should be multiplied by 7 and the second equation by 6.
12

The linear combination method is applied to a system of equations as shown.4(.25x + .5y = 3.75) → x + 2y = 15 (4x – 8y = 12) → x – 2y = 3 2x = 18

Question illustration
A
(1,2)
B
(3,9)
C
(5,5)
D
(9,3)
13

Which statement is true about the equations –3x + 4y = 12 and x – y = 1?

Question illustration
A
The system of the equations has exactly one solution at (–8, 3).
B
The system of the equations has exactly one solution at (–4, 3).
C
The system of the equations has no solution; the two lines are parallel.
D
The system of the equations has an infinite number of solutions represented by either equation.
14

The system of equations below has no solution.

Question illustration
A
Four-thirds x = 42
Option A
B
Option
Option B
C
Fifteen-halves y = 33
Option C
D
0 = 0
Option D
15

Which statement describes the graph of the system of equations below?1.5x + 0.2y = 2.681.6x + 0.3y = 2.98

A
The lines are parallel.
B
The lines overlap at all points.
C
The lines intersect at (1.6,1.4).
D
The lines intersect at (3.1,0.5).
16

Melinda and Paula shovel driveways and sidewalks in the winter as a way to earn extra money. Together they shoveled 450 square feet of sidewalk in 30 minutes. Then Melinda shoveled for 20 minutes while Paula shoveled for 25 minutes to complete 345 square feet of driveway.30x + 30y = 45020x + 25y = 345

A
3 square feet per minute
B
6 square feet per minute
C
9 square feet per minute
D
15 square feet per minute
17

The system of equations is solved using the linear combination method.What does 0 = −12 mean regarding the solution to the system?

Question illustration
A
There are no solutions to the system because the equations represent parallel lines.
B
There are no solutions to the system because the equations represent the same line.
C
There are infinitely many solutions to the system because the equations represent parallel lines.
D
There are infinitely many solutions to the system because the equations represent the same line.
18

To eliminate the y-terms and solve for x in the fewest steps, by which constants should the equations be multiplied by before adding the equations together?First Equation: 5x − 4y = 28Second equation: 3x - 9y = 30

A
The first equation should be multiplied by 3 and the second equation by 5.
B
The first equation should be multiplied by 3 and the second equation by −5.
C
The first equation should be multiplied by 9 and the second equation by 4.
D
The first equation should be multiplied by 9 and the second equation by −4.

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