Solving a System of Two Linear Equations by Using the Elimination Method or the Substitution Method

Why Do We Need Another Method? Why Isn't Graphing Good Enough?

Graphing to solve linear systems is a great way to see what is happening, and visualizing what you are doing in any algebra class is useful. With technology, using the graphing method can also be quite accurate, but without using technology graphing can be problematic, especially when the solution has coordinates that are fractions or decimals. We can't graph accurately by hand to get the right answer when we have non-integer coordinates in our solution.

So, please watch this video to learn how to solve systems of two linear equations using the Elimination Method:

The Elimination Method

One way to solve a system of two equations, called the elimination method, uses multiplication and addition to eliminate a variable from one equation. To eliminate a variable, the coefficients of that variable in the two equations must be additive inverses. To achieve this, we use properties of algebra to change the system to an equivalent system, one with the same solution set. The three transformations that produce an equivalent system are listed next.

  1. 1.Interchange any two equations of the system.

  2. 1.Multiply or divide any equation of the system by a nonzero real number.

  3. 1.Replace any equation of the system by the sum of that equation and a multiple of another equation in the system.

Here is another example of a problem solved using elimination:

Another way to solve a system of linear equations is the substitution method, explained below:

The Substitution Method

In a system of two equations with two variables, the substitution method involves using one equation to find an expression for one variable in terms of the other, and then substituting into the other equation of the system.

Here is another example using the substitution method: