Solution of a system of equations

Consider the following system of linear equations:

$2x-y=3$

$3x+2y=8$

In each of the above equations, let us substitute $x=2$ and $y=1$.

First equation:

$2\cdot 2-1=3$

Simplifying,

$3=3$

This is a true statement, since the left-hand side equals the right-hand side. Therefore, $x=2$ and $y=1$ are the solution of the first equation.

Second equation:

$3\cdot 2+2\cdot 1=8$

Simplifying,

$8=8$

This is also a true statement, since the left-hand side equals the right-hand side. Therefore, $x=2$ and $y=1$ are also the solution of the second equation.

Since $x=2$ and $y=1$ are the solution of both equations in the system, they are the solution of the system. We write the solution as an ordered pair, $(x,y)$ — the $x$ value first, then the $y$ value. Therefore, the solution of the system as an ordered pair is $(2,1)$.

Note that the solution of a system of equations is the solution of every equation in the system. If a solution of one equation is not a solution of the other, then it is not a solution of the system.

There are several methods for finding the solution of a system of equations, such as the graphical method, the substitution method, and the elimination method.

Graphical method →