Three equations#2

$2x-6y+z=-12$ $\color{#D61F06}(1)$

$3x+5y-3z=-17$ $\color{#D61F06}(2)$

$-4x-y+2z=22$ $\color{#D61F06}(3)$

I will eliminate $x$ from $\color{#D61F06}(1)$ and $\color{#D61F06}(2)$ . Then I will eliminate $x$ again from $\color{#D61F06}(2)$ and $\color{#D61F06}(3)$ .

$\{$ $\color{#D61F06}(1)$ $\times -3\}$ $+$ $\{$ $\color{#D61F06}(2)$ $\times 2\}$ $\large \implies$ $28y-9z=2$ $\color{#D61F06}(4)$

$\{$ $\color{#D61F06}(2)$ $\times 4\}$ $+$ $\{$ $\color{#D61F06}(3)$ $\times 3\}$ $\large \implies$ $17y-6z=-2$ $\color{#D61F06}(5)$

Now I will eliminate $z$ from $\color{#D61F06}(4)$ and $\color{#D61F06}(5)$ to solve for $y$ .

$\{$ $\color{#D61F06}(4)$ $\times -6\}$ $+$ $\{$ $\color{#D61F06}(5)$ $\times 9\}$ $\large \implies$ $y=2$

Substitute $y=2$ in $\color{#D61F06}(4)$ to solve for $z$

$28(2)-9z=2$

$56-2=9z$

$54=9z$

$6=z$

Substitute $z=6$ and $y=2$ in $\color{#D61F06}(1)$ to solve for $x$ .

$2x-6(2)+16=-12$

$2x=-12+12-6$

$2x=-6$

$x=-3$

Finally,

$x-y+z=-3-2+6=$ $\boxed{1}$

Given that,

$2x - 6y + z = -12$ .............(1)

$3x + 5y - 3z = -17$ ............(2)

$-4x - y + 2z = 22$ ..............(3)

Now, Multiply equation (1) by 3 and equation (2) by 1.

$6x - 18y + 3z = -36$ ........(4)

$3x + 5y - 3z = -17$ .........(2)

Add equations (4) and (2).

$9x - 13y = -53$ ................(5)

Multiply equation (1) by 2 and equation (3) by 1.

$4x - 12y + 2z = -24$ ..............(6)

$-4x - y + 2z = 22$ ....................(3)

Subtract equation (3) from equation (6).

$8x - 11y = -46$ .......(7)

Multiply equation (5) by 8 and equation (7) by 9.

$72x - 104y = -424$ .........(8)

$72x - 99y = -414$ ...........(9)

Subtract equation (9) from equation (8).

$-5y = -10 ⇒ y = 2$

Now we have the value of $y$ ,

Substitute it back into equation (7) to find $x$ .

$⇒8x - 22 = -46 ⇒ 8x = -24 ⇒ x = -3$

Now we have the values of $y$ and $x$ .

Substitute them back into equation (1) to find $z$

$⇒ -6 - 12 + z = -12 ⇒ z = 6$

Therefore, $⇒$ $x - y + z = -3 - 2 + 6 = 1$ .