When using matrices to solve a system of three equations in 3 variables x, y and z, write the system as an augmented matrix and use matrix row operations to get the matrix into row-echelon form. We use what is called Gaussian elimination.

When the matrix has 1's along the diagonal from upper left to lower row and 0's underneath the 1's, we can use the value obtained for z and substitute in for z in the second equation to solve for y and then substitute y and z in the first equation to solve for x.

Systems of equations may have none, one or infinitely many solutions.