Solving Systems of Equations by the Elimination Method

Most high school students find solving systems of equations very tricky, but understanding and using the elimination method is an excellent way to solve systems of equations faster and easier.

The elimination method involves adding or subtracting the equations in a system so that one variable is eliminated, leaving you with just one variable to solve for.

Read on to explore the steps involved in solving systems of equations by elimination method and learn some useful tips to make the process even simpler. Also, check out an article I wrote about solving systems of equations by substitution method.

What Is the Elimination Method?

The elimination method involves adding or subtracting equations to eliminate one of the variables, allowing us to solve for the remaining variable.

• If the coefficients of the two equations are opposite, you add the equations to eliminate a variable.
• If the coefficients of the two equations are equal, then you subtract the equations to eliminate a variable.

Even though most students find solving systems of equations by elimination method a little intimidating, it is actually a straightforward process that can be easily mastered with a bit of practice.

Here are the steps involved in using the elimination method to solve a system of equations.

Step 1: Rearrange the Equations

The first thing you need to do when solving a system of equations with the elimination method is to rearrange the equations so that the variables line up, which will make it easier to add or subtract the equations.

For example, if you have the following system of equations:

• 3y = 8 – 2x
• – 5y = 3 – 4x

You’ll want to rearrange them so they’re in the same order, like this:

• 2x + 3y = 8 (First equation)
• 4x – 5y = 3 (Second equation)

Step 2: Multiply One or Both Equations by a Number to Create Opposite Coefficients

The objective of this step is to create opposite coefficients for one of the variables in the two equations.

For example, in the system of equations above: We can eliminate the variable x by multiplying the first equation by -2 (2x + 3y = 8)*-2 = – 4x – 6y = – 16

This step is a crucial step in the elimination process.

• -4x – 6y = -16 (First equation)
• 4x – 5y = 3 (Second equation)

Step 3: Add or Subtract The Two Equations to Eliminate One Variable

The next step in the elimination method is to choose a variable to eliminate. In most cases, you will want to choose the variable that has the same coefficient in both equations.

In our example above, we will add the first equation to the second equation.

• This gives us the following:
• -4x – 6y = -16
• 4x – 5y = 3
• Adding these two equations together, we can eliminate the variable x and get:
• -y = -13
• Solving for y, we get y = 13.

Step 4: Substitute The Solution Back Into One of The Original Equations And Solve For The Remaining Variable

With one variable solved, we can now substitute it back into one of the original equations to solve for the other variable.

Using the first equation from our original system, we get the following:
2x + 3y = 8
2x + 3(13) = 8
2x + 39 = 8
2x = -31
x = -15.5
Therefore, our solution is x = -15.5 and y = 13.

It’s always a good idea to check your answer by plugging the values back into the original equations. If the values make both equations true, then you have the correct solution.

For example, checking our solution for the original system of equations, we get the following:
2(-15.5) + 3(13) = 8
-31 + 39 = 8
8 = 8
This equation is true, so our solution is correct.

Final Thoughts

Solving systems of equations by elimination method is a relatively simple process that involves five key steps:

1. Rearrange the equations
2. Create opposite coefficients
3. Eliminate one variable
4. Solve for the remaining variable