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
3y 
= 0 = 8 
5x  4y
y 
= 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 
Solution
In each equation, replace x with 5 and y with 2. Then simplify.

First equation 

Second equation 
Is
Is
Is 
3x
3(5)
15 
 
+ 
4y
4(2)
8
23 
= 23 = 23 ?
= 23 ?
= 23 ? Yes 
Is
Is 
x
(5) 
+ + 
y
(2)
3 
= 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).
