How to solve the system of equations by substitution

To solve systems using substitution, follow this procedure:

• Select one equation and solve it for one of its variables.
• In the other equation, substitute for the variable just solved.
• Solve the new equation.
• Substitute the value found into any equation involving both variables and solve for the other variable.
• Check the solution in both original equations.

Usually, when using the substitution method, one equation and one of the variables leads to a quick solution more readily than the other. That's illustrated by the selection of x and the second equation in the following example. 

Example 1

Solve this system of equations by using substitution.

Solve for x in the second equation. 

Substitute

Solve this new equation.

Substitute the value found for y into any equation involving both variables. 

Check the solution in both original equations.

The solution is x = 1, y = –2. 

If the substitution method produces a sentence that is always true, such as 0 = 0, then the system is dependent, and either original equation is a solution. If the substitution method produces a sentence that is always false, such as 0 = 5, then the system is inconsistent, and there is no solution.

What is the most useful technique for solving a system of equations?

Jenn, Founder Calcworkshop®, 15+ Years Experience (Licensed & Certified Teacher)

The Substitution Method!

Why?

Because it is used in such topics as nonlinear systems, linear algebra, computer programming, and so much more.

And the greatest thing about solving systems by substitution is that it’s easy to use!

The method of substitution involves three steps:

1. Solve one equation for one of the variables.
2. Substitute (plug-in) this expression into the other equation and solve.
3. Resubstitute the value into the original equation to find the corresponding variable.

Now at first glance, this may seem complicated, but I’ve got some helpful tricks for keeping things straight. In fact, we’re going to make a sort of circular circuit that helps to provide organization and efficiency to our method.

Using the Substitution Method to Solve

Remember, our goal when solving any system is to find the point of intersection. As we saw in our lesson titled the graphing method, we saw that some systems do not have solutions because they don’t intersect, and others coincide, which provides infinitely many solutions.

So when we solve systems by substitution, we will need to be on the lookout for these types of scenarios. If they are parallel and don’t intersect, then we are going to end up with an invalid answer, or as Purple Math calls it, a “garbage” result.

Together we will look at 11 examples of solving linear systems using the substitution method, and learn how to employ this technique for systems of two, three and even four equations.

Substitution Method (How-To) – Video

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