College Algebra, Chapter 7, 7.1, Section 7.1, Problem 30

State whether the system of linear equations $\left\{ \begin{array}{ccccc} x & & + 3z & = & 3 \\ 2x & + y & - 2z & = & 5 \\ & - y & + 8z & = & 8 \end{array} \right.$ is inconsistent or dependent. If it is dependent, find the complete solution.

We transform the system into row-echelon form.

$\left[ \begin{array}{cccc} 1 & 0 & 3 & 3 \\ 2 & 1 & -2 & 5 \\ 0 & -1 & 8 & 8 \end{array} \right]$

$R_2 - 2R_1 \to R_2$

$\left[ \begin{array}{cccc} 1 & 0 & 3 & 3 \\ 0 & 1 & -8 & -1 \\ 0 & -1 & 8 & 8 \end{array} \right]$

$R_3 + R_2 \to R_3$

$\left[ \begin{array}{cccc} 1 & 0 & 3 & 3 \\ 0 & 1 & -8 & -1 \\ 0 & 0 & 0 & 7 \end{array} \right]$

$\displaystyle \frac{1}{7} R_3$

$\left[ \begin{array}{cccc} 1 & 0 & 3 & 3 \\ 0 & 1 & -8 & -1 \\ 0 & 0 & 0 & 1 \end{array} \right]$

This last matrix is in row-echelon form, so we can stop the Gaussian Elimination process. Now if we translate the last row back into equation form, we get $0x + 0y + 0z = 1$, or $0 = 1$, which is false. This means that the system is inconsistent.