Answer
The system of equations is inconsistent and has no solutions.
Work Step by Step
We need to solve the given system of equations: $$2x -3y -z =0 ~~~(1) \\ -x +2y+z =5 ~~~(2) \\ 3x -4y-z=1 ~~~(3) $$
Add equations $(1)$ and $(2)$ to eliminate $z$. $$2x-3y -z- x+2y+z=0+5 \\ x-y=5 ~~~ (4)$$
We can notice that $z$ is eliminated.
Next, add equations $(2)$ and $(3)$ to eliminate $z$: .
$$ -x +2y+z+3x -4y-z=5+1\\ 2x-2y=6 \\ x-y=3 ~~~(5)$$
Now, subtract equations $(4)$ from equation $(5):$:
$x-y-x+y=5-3 \\ 0=2$
Since $0$ never equals $2$, the system of equations is inconsistent and has no solutions.
