# Systems of Linear Equations and Inequalities

Systems of linear equations and inequalities are a fundamental concept in algebra that involve finding the solutions to two or more equations or inequalities simultaneously. In other words, they involve finding the values of variables that satisfy multiple equations or inequalities at the same time. These systems can be solved using a variety of methods, including graphing, substitution, and elimination.

On the SAT Math section, questions involving systems of linear equations and inequalities typically test a student's ability to solve systems of equations and inequalities, and to apply this knowledge to real-world situations. Students may be asked to find the solutions to systems of equations or inequalities, interpret the meaning of those solutions in context, or identify which of several graphs represents the solutions to a given system.

Practice is key when it comes to mastering systems of linear equations and inequalities, so offering students the opportunity to try your practice test is a great way to help them improve their skills and build confidence before taking the SAT. As always, should you have any trouble with the questions on this test, don't hesitate to reach out for personalized SAT tutoring to help you ace this section.

Good luck!

## Question 1:

Solve the following system of linear equations:

x + y = 7

2x - y = 1

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1. x = 2, y = 5

2. x = 3, y = 4

3. x = 4, y = 3

4. x = 5, y = 2

## Question 2:

Which of the following systems of linear equations has infinitely many solutions?

1. 3x + 2y = 6  |  6x + 4y = 12

2. x - 2y = 5  |  x + y = 3

3. 2x + 3y = 7  |  4x - 6y = 2

4. y = 2x + 1  |  y = -x - 1

## Question 3:

What is the solution to the system of inequalities:

1. y ≤ 3x - 2

2. y > -2x + 5

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1. The region above y = 3x - 2 and below y = -2x + 5

2. The region below y = 3x - 2 and above y = -2x + 5

3. The region above y = 3x - 2 and to the right of y = -2x + 5

4. The region below y = 3x - 2 and to the left of y = -2x + 5

## Question 4:

If the system of linear equations below has no solution, what is the value of k?

2x + 3y = 5

4x + ky = 10

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1. 6

2. 8

3. 9

4. 10

## Question 5:

Which of the following systems of linear inequalities is represented by the graph described below?

Graph: [Imagine a graph with x and y axes. The region between the lines y = x - 1 and y = -x + 3 is shaded.]

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1. y ≥ x - 1 and y ≤ -x + 3

2. y ≤ x - 1 and y ≥ -x + 3

3. y ≥ x - 1 and y ≥ -x + 3

4. y ≤ x - 1 and y ≤ -x + 3

## Question 6:

Solve the following system of linear equations using the elimination method:

3x - 2y = 8

6x + 5y = 1

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1. x = -1, y = 2

2. x = 1, y = -1

3. x = 2, y = -1

4. x = -2, y = 1

## Question 7:

Which of the following systems of linear equations has exactly one solution?

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1. x - y = 3  |  2x - 2y = 6

2. 4x + 6y = 12  |  2x + 3y = 6

3. x + 2y = 5  |  3x - y = 4

4. 3x - 4y = 8  |  -9x + 12y = -24

## Question 8:

What is the solution to the system of inequalities:

2x + y ≥ 6

x - y ≤ 2

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1. The region above y = -2x + 6 and below y = x + 2

2. The region below y = -2x + 6 and above y = x + 2

3. The region above y = 2x + 6 and below y = x - 2

4. The region below y = 2x + 6 and above y = x - 2

## Question 9:

If the system of linear equations below has an infinite number of solutions, what is the value of k?

5x - 3y = 10

kx - 6y = 20

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1. 2

2. 2.5

3. 10

4. 12

## Question 10:

Which of the following systems of linear inequalities is represented by the graph described below?

Graph: [Imagine a graph with x and y axes. The region below the lines y = 2x + 4 and y = -x + 6 is shaded.]

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1. y ≤ 2x + 4 and y ≤ -x + 6

2. y ≥ 2x + 4 and y ≥ -x + 6

3. y ≤ 2x + 4 and y ≥ -x + 6

4. y ≥ 2x + 4 and y ≤ -x + 6