We can solve a system of linear simultaneous equations by forming the augmented matrix and using elementary row operations to reduce to upper triangular form. The solutions can then be found by back substitution.

Example: Solve the system of equations

2x-3y+5z=17

x-2y-3z=-3

3x+y+z=18

The augmented matrix is

We can interchange rows 1 and 2, to make the entry in the upper left hand corner 1.

Subtract twice the first row from the second row, and three times the first row from the second row.

Subtract seven times the second row from the third row.

From the third eow,

From the second row,

From the first row,