Question

In Problems 37–72, solve each system of equations using matrices (row operations). If the system has no solution, say that it is inconsistent.

$\left\{\begin{array}{l}2x-2y+3z=6\\ -x+2y+z=5\\ 3x-4y-z=1\end{array}\right.$

Short answer

The system of equation is inconsistent.

Step-by-step solution

Step 1. Given information. 

The given system of equation are

$\left\{\begin{array}{l}2x-2y+3z=6\\ -x+2y+z=5\\ 3x-4y-z=1\end{array}\right.$

Step 2. Calculation. 

The augmented matrix of the system is:

$\left[\begin{array}{cc}\begin{array}{ccc}2& -2& 3\\ -1& 2& z\\ 3& -4& -1\end{array}& \begin{array}{c}6\\ 5\\ 1\end{array}\end{array}\right]$

Perform the row operations $R_2=r_2-2r_1;R_3=r_3+r_1$:

$\left[\begin{array}{cc}\begin{array}{ccc}2& -2& 3\\ 0& 1& -4\\ 0& 1& -4\end{array}& \begin{array}{c}6\\ -12\\ 7\end{array}\end{array}\right]$

Perform the row operations $R_3=r_3-r_2$:

$\left[\begin{array}{cc}\begin{array}{ccc}2& -2& 3\\ 0& 1& -4\\ 0& 0& 0\end{array}& \begin{array}{c}6\\ -12\\ 19\end{array}\end{array}\right]$

Use the obtained matrix to write the system of equations.

$$\begin{gathered} 2x-2y+3z=6 \\ y-4z=-12 \\ 0=19 \end{gathered}$$

This system is impossible to solve. Therefore, this system of equation is inconsistent.

Step 3. Conclusion.

The system of equation is inconsistent.