# Solving systems of equations in three variables

When solving systems of equation with three variables we use the elimination method or the substitution method to make a system of two equations in two variables.

**Example**

Solve the systems of equations (this example is also shown in our video lesson)

$$\left\{\begin{matrix} x+2y-z=4\\ 2x+y+z=-2\\ x+2y+z=2 \end{matrix}\right.$$

First we add the first and second equation to make an equation with two variables, second we subtract the third equation from the second in order to get another equation with two variables. Now we have a system of two equations with two variables:

$$\left\{\begin{matrix} 3x+3y = 2\\ x-y=-4 \end{matrix}\right.$$

We then multiply the second equation with 3 on both sides and add that to the first equation:

$$6x=-10$$

$$x=\frac{-10}{6}$$

We plug this value into the 3x+3y=2 equation in order to determine our y-value:

$$\begin{array}{lcl} \\ 3\cdot \frac{-10}{6}+3y&=&2\\ \\ -5+3y&=&2\\ 3y&=&7\\ \\ y&=&\frac{7}{3}\\ \end{array}$$

Last we plug our x- and y-value into any equation in first system in order to determine our z-value:

$$\begin{array}{lcl} x+2y-z&=&4\\ \frac{-10}{6}+2\cdot \frac{7}{3}+z&=&2\\ 3+z&=&2\\ z&=&-1\\ \end{array}$$

**Video lesson**

Solve the systems of equation in our example.