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The linear system has the matrix formThe first equation gives $3x_2+2x_3=-4$. Substitute it into the second to obtain $x_1=2+4x_2+2x_3=x_2-2$. Substitution into the third gives $x_2+x_3=-7$. ConsequentlyIt follows thatSubstitution verifies all three equations. Also $\det A=1\ne 0$, so this solution of a linear system is unique.

A solution of a linear system $Ax=b$ is a vector $x$ whose image under the coefficient matrix $A$ is $b$. Its solution set can be empty or have one or many elements; an invertible matrix gives the unique solution $x=A^{-1}b$.

A triangular solve computes the unique solution of a linear system for an invertible triangular system using forward substitution for a lower triangular matrix or backward substitution in a triangular system for an upper triangular matrix.