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Systems of Linear Equations

A system of equations consists of two or more equations, each of which contains at least one variable. Here are three examples:

System 1 System 2 System 3

3x + y

4x - 2y

= -5

= 7

-7x + 9y


= 0

= 8

5x - 4y


= 11

= 3x + 1


Each system is called a linear system in two variables. This is because the graph of each equation is linear (that is, the graph is a straight line) and two variables are involved.

An ordered pair, (x, y), is a solution of a linear system of equations in two variables if the ordered pair makes each equation true. An ordered pair that is a solution is said to satisfy the system.


Example 1

Determine if (5, -2) is a solution of this system.

3x - 4y

x + y

= 23

= 3

First equation

Second equation



In each equation, replace x with 5 and y with -2. Then simplify.

  First equation   Second equation














= 23

= 23 ?

= 23 ?

= 23 ? Yes











= 3

= 3 ?

= 3 ? Yes


Since (5, -2) satisfies each equation, it is a solution of the system.

The solution can be written as x = 5 and y = -2, or simply (5, -2).

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