Question

Solve each system of equations. If the system has no solution, say that it is inconsistent.

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

Short answer

The system of equation is inconsistent

Step-by-step solution

Given information

We are given a system of equation

$$\begin{gathered} 3x-2y+2z=6\left(1\right) \\ 7x-3y+2z=-1\left(2\right) \\ 2x-3y+4z=0\left(3\right) \end{gathered}$$

Add equation 1 and 2

We get,

$3x-2y+2z=6-7x+3y-2z=1-4x+y=7$

We get,$-4x+y=7\left(4\right)$

Now multiply equation 2 by 2 and subtract from equation 3

We get,

$$\begin{gathered} 2(7x-3y+2z=-1) \\ 14x-6y+4z=-2 \end{gathered}$$

Now we subtract

$14x-6y+4z=-2+-2x+3y-4z=012x-3=-2$

We get,

$$\begin{gathered} 12x-3y=-2 \\ -4x+y=\frac{2}{3} \end{gathered}$$

But from previous step we have $-4x+y=7$

As both equations cannot have different values The system of equation is inconsistent

Conclusion

The system of equation is inconsistent