Answer
The solution is $x = 2$, $y = -1$, and $z = 5$ or $(2, -1, 5)$.
Work Step by Step
Use the expression given for $z$ to substitute into the first and second equations:
$2x + y + (2x - y) = 8$
$x + 2y - (2x - y) = -5$
Distribute first:
$2x + y + 2x - y = 8$
$x + 2y - 2x + y = -5$
Group like terms:
$(2x + 2x) + (y - y) = 8$
$(x - 2x) + (2y + y) = -5$
Combine like terms:
$4x + 0 = 8$
$-x + 3y = -5$
Solve the first equation for $x$ by dividing both sides of the equation by $4$:
$x = 2$
Plug in this value for $x$ into the second equation:
$-2 + 3y = -5$
Add $2$ to each side of the equation:
$3y = -3$
Divide each side by $3$ to solve for $y$:
$y = -1$
Now that we have the values for $x$ and $y$, we can substitute these values in for $x$ and $y$ into the second equation to solve for $z$:
$2 + 2(-1) - z = -5$
Multiply first:
$2 - 2 - z = -5$
Add to simplify:
$0 - z = -5$
Divide each side of the equation by $-1$:
$z = 5$
The solution is $x = 2$, $y = -1$, and $z = 5$ or $(2, -1, 5)$.
