Answer
The solution is $x = 1$, $y = 3$, and $z = 2$ or $(1, 3, 2)$
Work Step by Step
Set up a system of equations consisting of the first and third equations so that we can eliminate at least one of the variables:
$x - y + z = 0$
$-x + y - 2z = -2$
Combining the two equations by adding them together:
$-z = -2$
Divide both sides of the equation by $-1$:
$z = 2$
Now, let's set up a system of equations that eliminates another variable, keeping the $z$ terms. Let's use the second and third equations:
$3x - 2y + 6z = 9$
$-x + y - 2z = -2$
Let's eliminate the $y$ term by multiplying the second equation by $2$:
$3x - 2y + 6z = 9$
$-2x + 2y - 4z = -4$
Combine the two equations by adding them together:
$x + 2z = 5$
Plug in the value we found for $z$ to solve for $x$:
$x + 2(2) = 5$
Multiply first:
$x + 4 = 5$
Subtract $4$ from each side of the equation:
$x = 1$
Now that we have values for both $x$ and $z$, we can plug them into one of the original equations to find $y$. Let's use the first equation:
$1 - y + 2 = 0$
Add like terms:
$-y + 3 = 0$
Subtract $3$ from each side of the equation:
$-y = -3$
Divide each side of the equation by $-1$:
$y = 3$
Our solution is $x = 1$, $y = 3$, and $z = 2$.
