Linear Equations: Solutions Using Matrices with Three Variables

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.

equation

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

equation

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

Put the equation in matrix form.

equation

Eliminate the x‐coefficient below row 1.

equation

Eliminate the y‐coefficient below row 5.

equation

Reinserting the variables, this system is now equation

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.

equation

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.

equation

Put the equations in matrix form.

equation

Eliminate the x‐coefficient below row 1.

equation

Eliminate the y‐coefficient below row 5.

equation

Reinserting the variables, the system is now: equation

Equation (9) can be solved for z.

equation

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

equation

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

equation

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