Q. 56

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

$\{\begin{cases} 3 x - 2 y + 2 z = 6 \\ 7 x - 3 y + 2 z = - 1 \\ 2 x - 3 y + 4 z = 0 \end{cases}$

Verified

Answer

The system of equation is inconsistent.

1Step 1. Given information.

The given system of equation are

$\{\begin{cases} 3 x - 2 y + 2 z = 6 \\ 7 x - 3 y + 2 z = - 1 \\ 2 x - 3 y + 4 z = 0 \end{cases}$

2Step 2. Calculation.

The augmented matrix of the system is:

$[\begin{matrix} \begin{matrix} 3 & - 2 & 2 \\ 7 & - 3 & 2 \\ 2 & - 3 & 4 \end{matrix} & \begin{matrix} 6 \\ - 1 \\ 0 \end{matrix} \end{matrix}]$

Perform the row operations $R_{2} = 3 r_{2} - 7 r_{1}, R_{3} = 3 r_{3} - 2 r_{1}$ :

$[\begin{matrix} \begin{matrix} 3 & - 2 & 2 \\ 0 & 5 & - 8 \\ 0 & - 5 & 8 \end{matrix} & \begin{matrix} 6 \\ - 45 \\ 12 \end{matrix} \end{matrix}]$

Perform the row operations $R_{3} = r_{3} + r_{2}$ :

$[\begin{matrix} \begin{matrix} 3 & - 2 & 2 \\ 0 & 5 & - 8 \\ 0 & 0 & 0 \end{matrix} & \begin{matrix} 6 \\ - 45 \\ - 57 \end{matrix} \end{matrix}]$

Use the obtained matrix to write the system of equations.

$3 x - 2 y + 2 z = 6 5 y - 8 z = - 45 0 = - 57$

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

3Step 3. Conclusion.

The system of equation is inconsistent.