Q. 51

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

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

Verified

Answer

The system of equation is inconsistent.

1Step 1. Given information.

The given system of equation are

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

2Step 2. Calculation.

The augmented matrix of the system is:

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

Perform the row operations $R_{1} = \frac{1}{2} r_{1}$ :

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

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

$[\begin{matrix} \begin{matrix} 1 & - 1 & - 1 \\ 0 & 5 & 3 \\ 0 & 5 & 3 \end{matrix} & \begin{matrix} 1 \\ 0 \\ - 3 \end{matrix} \end{matrix}]$

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

$[\begin{matrix} \begin{matrix} 1 & - 1 & - 1 \\ 0 & 5 & 3 \\ 0 & 0 & 0 \end{matrix} & \begin{matrix} 1 \\ 0 \\ - 3 \end{matrix} \end{matrix}]$

Use the obtained matrix to write the system of equations.

$x - y - z = 1 5 y + 3 z = 0 0 = - 3$

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

3Step 3. Conclusion.

The system of equation is inconsistent.