## Solve Systems by Elimination

### Learning Outcomes

• Solve systems of equations by addition
• Express the solution of a system of dependent equations containing two variables

A third method of solving systems of linear equations is the elimination method. In this method, we add two terms with the same variable, but opposite coefficients, so that the sum is zero. Of course, not all systems are set up with the two terms of one variable having opposite coefficients. Often we must adjust one or both of the equations by multiplication so that one variable will be eliminated by elimination.

### Example

Solve the given system of equations by elimination.

$\begin{array}{l}x+2y=-1\hfill \\ -x+y=3\hfill \end{array}$

In the following video, you will see another example of how to use the method of elimination to solve a system of linear equations.

Sometimes we have to do a couple of steps of algebra before we can eliminate a variable from a system and solve it. In the next example, you will see a technique where we multiply one of the equations in the system by a number that will allow us to eliminate one of the variables.

### Example

Solve the given system of equations by the elimination method.

$\begin{array}{l}3x+5y=-11\hfill \\ x - 2y=11\hfill \end{array}$

Below is another video example of using the elimination method to solve a system of linear equations.

In the next example, we will see that sometimes both equations need to be multiplied by different numbers in order for one variable to be eliminated.

### Example

Solve the given system of equations in two variables by elimination.

$\begin{array}{c}2x+3y=-16\\ 5x - 10y=30\end{array}$

Below is a summary of the general steps for using the elimination method to solve a system of equations.

### How To: Given a system of equations, solve using the elimination method

1. Write both equations with and y-variables on the left side of the equal sign and constants on the right.
2. Write one equation above the other, lining up corresponding variables. If one of the variables in the top equation has the opposite coefficient of the same variable in the bottom equation, add the equations together, eliminating one variable. If not, use multiplication by a nonzero number so that one of the variables in the top equation has the opposite coefficient of the same variable in the bottom equation, then add the equations to eliminate the variable.
3. Solve the resulting equation for the remaining variable.
4. Substitute that value into one of the original equations and solve for the second variable.
5. Check the solution by substituting the values into the other equation.

In the next example, we will show how to solve a system with fractions. As with single linear equations, the easiest way to solve is to clear the fractions first with the least common denominator.

### Example

Solve the given system of equations in two variables by elimination.

$\begin{array}{l}\dfrac{x}{3}+\dfrac{y}{6}=3\hfill \\ \dfrac{x}{2}-\dfrac{y}{4}=\text{ }1\hfill \end{array}$

In the following video, you will find one more example of using the elimination method to solve a system; this one has coefficients that are fractions.

Recall that a dependent system of equations in two variables is a system in which the two equations represent the same line. Dependent systems have an infinite number of solutions because all of the points on one line are also on the other line. After using substitution or elimination, the resulting equation will be an identity such as $0=0$. The last example includes two equations that represent the same line and are therefore dependent.

### Example

Find a solution to the system of equations using the elimination method.

$\begin{array}{c}x+3y=2\\ 3x+9y=6\end{array}$

In the following video, we show another example of solving a system that is dependent using elimination.

In our last video example, we present a system that is inconsistent; it has no solutions which means the lines the equations represent are parallel to each other.