To solve a system of linear equations using matrices, one must systematically transform the initial augmented matrix into row-echelon form. Begin by writing the augmented matrix representing the system. Then, apply row operations to force the entry in row $1$, column $1$ to be $1$, followed by getting zeros in the remainder of column $1$ below that $1$. Continue this structured process—for instance, forcing the entry in row $2$, column $2$ to be $1$—until the entire matrix reaches row-echelon form. Finally, translate the matrix back into a system of equations, use back-substitution to determine any remaining variables, express the final solution as an ordered pair or triple, and verify the result against the original equations.