Recall that we worked with systems of linear equations such as the following:

3x + 4y = -6

5x – 2y = 1

5x + 6y – z = 10

7y + 4z = 3
z = -1

We can use a matrix to rewrite these systems in a simpler way. We use what is called an augmented matrix, which has a vertical bar separating the columns of the matrix. The group on the left of the bar are the coefficients of each variable and the group to the right of the bar are the constants (the numbers after the = in each equation in the system). In the systems above, the augmented matrices are as follows:

To solve a system of linear equations in three variables, we wish to produce a matrix with 1's along the diagnoal from the upper left to the lower right of the matrix, with 0's undereath the 1's. Such a matrix will look as follows:

The letters a through f represent real numbers. Recall that the elements of the augemented matrix to the left of the vertical bar represent the coefficients of the variables. Therefore in the above augemented matrix, we can conclude that

x + ay + bz = c

y + dz = e
z = f

Since we know the value of z, we can subsitute that into y + dz = e to solve for y. Then we can substitute the values for y and z into x + ay + bz = c and solve for x.