Answer
Our solution is $x = 6$, $y = 0$, and $z = -2$ or $(6, 0, -2)$
Work Step by Step
Use the expression given for $x$ to substitute into the first two equations:
$3(-3z) + y - 2z = 22$
$-3z + 5y + z = 4$
Group like terms:
$y + (-9z - 2z) = 22$
$5y + (-3z + z) = 4$
Combine like terms:
$y + (-11z) = 22$
$5y + (-2z) = 4$
Simplify:
$y - 11z = 22$
$5y - 2z = 4$
We want to modify these equations such that one variable in both equations is the same but differing in sign. We can accomplish this by multiplying the first equation by $-5$:
$-5y + 55z = -110$
$5y - 2z = 4$
Combine the two equations by adding them together:
$53z = -106$
Divide both sides of the equation by $53$:
$z = -2$
Plug this value for $z$ into one of the modified equations to find $y$:
$5y - 2(-2) = 4$
Multiply to simplify:
$5y + 4 = 4$
Subtract $4$ on both sides of the equation:
$5y = 0$
Divide both sides by $5$:
$y = 0$
Now that we have values for both $y$ and $z$, we can plug them into one of the original equations to find the value of $x$. Let's use the second equation:
$x + 5(0) + (-2) = 4$
Multiply to simplify:
$x + 0 - 2 = 4$
Combine like terms:
$x - 2 = 4$
Add $2$ to each side of the equation:
$x = 6$
Our solution is $x = 6$, $y = 0$, and $z = -2$.
