Question

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

$$\begin{gathered} 2x-3y-z=0 \\ -x+2y+z=5 \\ 3x-4y-z=1 \end{gathered}$$

Short answer

The given system of equation is inconsistent

Step-by-step solution

Given information

We are given a system of equation

$$\begin{gathered} 2x-3y-z=0\left(1\right) \\ -x+2y+z=5\left(2\right) \\ 3x-4y-z=1\left(3\right) \end{gathered}$$

Multiply equation 2 by 2 and subtract from equation 1 we get,

We get,

$$\begin{gathered} 2(-x+2y+z=5) \\ -2x+4y+2z=10 \end{gathered}$$

Also

$-2x+4y+2z=10+2x-3y-z=0y+z=10$

We get$y+z=10\left(4\right)$

Multiply equation 2 by 3 and subtract from equation 3

We get,

$$\begin{gathered} 3(-x+2y+z=5) \\ -3x+6y+3z=15 \end{gathered}$$

Also

$-3x+6y+3z=15+3x-4y-z=12y+2z=16$

We get

$$\begin{gathered} 2y+2z=16 \\ y+z=8\left(5\right) \end{gathered}$$

But from 4 we have $y+z=10$

Same equations cannot have different values

Hence the system is inconsistent

Conclusion

The given system of equation is inconsistent