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Systems of Equations

SAT Math Overview. Topic: Systems of Equations

Overview

Expect to see some questions on the topic of Linear Systems on the SAT. In this section, we will review the following aspects of this topic:

  • Definition of a Linear System
  • Solving a Linear System
    • Substitution Method
    • Elimination Method
    • Graphical Method

Let’s look at each of these in more detail:

Definition of a Linear System

A linear system involves two or more linear equations that represent graphs of linear functions. Where the graphs intersect is the solution to the system.

Linear System. A collection of linear equations, where the number of equations matches the number of variables, allowing for an algebraic solution.

The kind of linear systems questions you will find on the SAT involve two variables. Here is an example of a linear system with the solution highlighted.

A graphical solution to a linear system showing the intersection of the graphs of two lines.

If a linear system doesn’t have a solution, that means the graphs of the lines are parallel. 

The graph of a linear system with no solution because the graphs of the lines are parallel.

Lines that are parallel have the same slope.

To learn more about the basics of linear systems, click on this link. 

Solving a System Using the Substitution Method

In this section we go over the process of solving a linear system using the Substitution Method. This method is summarized below.

Solving a Linear System Using Substitution. Taking the value for y from one linear equation and substituting it into the second equation to solve for x. From this the solution to the linear system is found.

 

SAT Skill: Solving Systems Using Substitution

Example 1

What is the solution to this system of equations?

A linear system consisting of y equals 7 x plus 10and y equals 4 x minus 5

Substitute y in one of the equations with the corresponding expression with x. Then solve for x.

Solving the linear system consisting of y equals 7 x plus 10 and y equals 4 x minus 5

Plug in the value for x to find y:

Solving the linear system consisting of y equals 7 x plus 10 and y equals 4 x minus 5

The solution to the system is (-5, -25).

Example 2

What is the solution to this system of equations?

A linear system consisting of y minus 2 x equals negative 8 and y plus 5 x equals negative 1

Rewrite one equation to get a substitution value for y and plug that into the second equation. Then solve for x: 

Solution to a linear system consisting of y minus 2 x equals negative 8 and y plus 5 x equals negative 1

Plug in the value for x to find y:

Solution to a linear system consisting of y minus 2 x equals negative 8 and y plus 5 x equals negative 1

The solution to the system is (1, -6).

 

To see examples of solving systems using the Substitution Method, click on this link.

Solving a Linear System Using the Elimination Method

In this section we go over the process of solving a linear system using the Elimination Method. This method is summarized below.

Solving a Linear System Using Elimination. Combining the equations of a linear system in such a way as to remove enough variables to solve the system.

 

SAT Skill: Solving Systems Using Elimination

Example 1

What is the solution to this system of equations?

A linear system consisting of 2 y equals 2 x plus 12 and y equals negative 4 x minus 4

Multiply one of the equations by a factor such that when both equations are added, one of the variables is eliminated. Solve for the remaining variable.

Solution to a linear system consisting of 2 y equals 2 x plus 12 and y equals negative 4 x minus 4

Plug in the value for x to find y:

Solution to a linear system consisting of 2 y equals 2 x plus 12 and y equals negative 4 x minus 4

The solution to the system is (-2, 4).

 

To see examples of solving systems using the Elimination Method, click on this link.

Solving a System Using a Graphical Approach

In this section we go over the process of solving a linear system using the Graphical Method. This method is summarized below.

Graphical Solution to a Linear System. Where the graphs of a linear system intersect is the solution to the linear system. If there is no intersection, there is no solution.


 

SAT Skill: Solving Systems Using the Graphical Method

Example 1

What is the solution to this system of equations?

A linear system consisting of y equals negative 8 x plus 9 and y equals negative 6 x plus 3

Graph the linear functions and find the point of intersection:

A graphical solution to the linear system consisting of y equals negative 8 x plus 9 and y equals negative 6 x plus 3

The solution to the system is (3, -15).

 

To see examples of solving linear systems by graphing, click on this link. 

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