The elimination method for solving systems of equations, also known as elimination by addition, is a way to eliminate one of the variables in the system in order to more simply evaluate the remaining variable. Once the values for the remaining variables have been found successfully, they are substituted into the original equation in order to find the correct value for the other variable.

The elimination method follows these steps:

1. Rewrite the equations so the variables line up.
2. Modify one equation so both equations have a variable that will cancel itself out when the equations are added together.
3. Add the equations and eliminate the variable.
4. Solve for the remaining variable.
5. Back-substitute and solve for the other variable.

Solving with the Elimination Method

The elimination method can be demonstrated by using a simple example:

$\displaystyle 4x+y=8 \\ 2y+x=9$

First, line up the variables so that the equations can be easily added together in a later step:

$\displaystyle \begin{aligned} 4x+y&=8 \\x+2y&=9 \end{aligned}$

Next, look to see if any of the variables are already set up in such a way that adding them together will cancel them out of the system. If not, multiply one equation by a number that allow the variables to cancel out. In this example, the variable y can be eliminated if we multiply the top equation by $-2$ and then add the equations together.

Multiplication step:

$\displaystyle \begin{aligned} -2(4x+y&=8) \\x+2y&=9 \end{aligned}$

Result:

$\displaystyle \begin{aligned} -8x-2y&=-16 \\x+2y&=9 \end{aligned}$

Now add the equations to eliminate the variable y.

$\displaystyle \begin{aligned} -8x+x-2y+2y&=-16+9 \\-7x&=-7 \end{aligned}$

Finally, solve for the variable x.

$\displaystyle \begin{aligned} -7x&=-7 \\x&=\frac{-7}{-7} \\x&=1 \end{aligned}$

Then go back to one of the original equations and substitute the value we found for x. It is easiest to pick the simplest equation, but either equation will work.

$\displaystyle \begin{aligned} 4x+y&=8 \\4(1)+y&=8 \\4+y&=8 \\y&=4 \end{aligned}$ 

Therefore, the solution of the equation is (1,4). It is always important to check the answer by substituting both of these values in for their respective variables into one of the equations.

$\displaystyle \begin{aligned} 4x+y&=8 \\4(1)+4&=8 \\4+4&=8 \\8&=8 \end{aligned} $