Solving a system of equations by using matrices is merely an organized manner of using the elimination method.

##### Example 1

Solve this system of equations by using matrices.

The goal is to arrive at a matrix of the following form.

To do this, you use row multiplications, row additions, or row switching, as shown in the following.

Put the equation in matrix form.

Eliminate the *x*‐coefficient below row 1.

Eliminate the *y*‐coefficient below row 5.

Reinserting the variables, this system is now

Equation (9) now can be solved for *z*. That result is substituted into equation (8), which is then solved for *y*. The values for *z* and *y* then are substituted into equation (7), which then is solved for *x*.

The check is left to you. The solution is *x* = 2, *y* = 1, *z* = 3.

##### Example 2

Solve the following system of equations, using matrices.

Put the equations in matrix form.

Eliminate the *x*‐coefficient below row 1.

Eliminate the *y‐*coefficient below row 5.

Reinserting the variables, the system is now:

Equation (9) can be solved for *z.*

Substitute into equation (8) and solve for *y*.

Substitute into equation (7) and solve for *x*.

The check of the solution is left to you. The solution is , , .