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.

$\left\{\begin{array}{l}3x-2y+2z=6\\ 7x-3y+2z=-1\\ 2x-3y+4z=0\end{array}\right.$

Short answer

The system of equation is inconsistent.

Step-by-step solution

Step 1. Given information. 

The given system of equation are

$\left\{\begin{array}{l}3x-2y+2z=6\\ 7x-3y+2z=-1\\ 2x-3y+4z=0\end{array}\right.$

Step 2. Calculation. 

The augmented matrix of the system is:

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

Perform the row operations $R_2=3r_2-7r_1,R_3=3r_3-2r_1$:

$\left[\begin{array}{cc}\begin{array}{ccc}3& -2& 2\\ 0& 5& -8\\ 0& -5& 8\end{array}& \begin{array}{c}6\\ -45\\ 12\end{array}\end{array}\right]$

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

$\left[\begin{array}{cc}\begin{array}{ccc}3& -2& 2\\ 0& 5& -8\\ 0& 0& 0\end{array}& \begin{array}{c}6\\ -45\\ -57\end{array}\end{array}\right]$

Use the obtained matrix to write the system of equations.

$$\begin{gathered} 3x-2y+2z=6 \\ 5y-8z=-45 \\ 0=-57 \end{gathered}$$

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

Step 3. Conclusion. 

The system of equation is inconsistent.