Answer
$z = 2$
Work Step by Step
We want to find the value for $z$ from these three equations.
We can solve this system of equations using the elimination method.
We see that for the second and third equations, the $y$ terms are the same except for the sign, so if we add these two equations together, we will get rid of the $y$ term and have only two variables left, so let's go ahead and do that:
$-x + y - 2z = -5$
$3x - y - 3z = -7$
Let's get rid of the $y$ term by adding these two equations to obtain:
$2x - 5z = -12$ (Let us call this equation $4$.)
Now, let's turn our attention to the first and second equations. Let's convert the second equation so that the $y$ terms of these two equations are exactly the same but differing in sign.
We do this by multiplying the second equation by $3$:
$3(-x + y - 2z) = 3(-5)$
Distribute and multiply to obtain:
$-3x + 3y - 6z = -15$
Pair this equation with the first equation in the system of equations:
$-3x + 3y - 6z = -15$
$ 2x - 3y + z = 6$
Add the equations together to obtain:
$-x - 5z = -9$ (Let us call this Equation $5$.)
Now, we can add equations $4$ and $5$ together to eliminate one more variable.
Let's eliminate the $x$ value because the calculations would be less complex.
Multiply Equation $5$ by $2$, and keep Equation $4$ as-is:
$ 2(-x - 5z) = 2(-9)$
Let's distribute and multiply to simplify:
$-2x - 10z = -18$
Now, we can pair this equation with Equation $4$ to have:
$2x - 5z = -12$
$-2x - 10z = -18$
Add the two equations together to eliminate the $x$ terms:
$-15z = -30$
Divide each side by $-15$ to solve for $z$:
$z = 2$
