Answer
$(0, 2, -3)$
Work Step by Step
Label the original equations first:
1. $x - y - 2z = 4$
2. $-x + 2y + z = 1$
3. $-x + y - 3z = 11$
The first step is to combine equations $1$ and $3$ to see if one variable can be eliminated:
1. $x - y - 2z = 4$
3. $-x + y - 3z = 11$
Add the equations together:
$-5z = 15$
Divide both sides of the equation by $-5$ to solve for $z$:
$z = -3$
Substitute this value for $z$ into both equations $1$ and $2$ to eliminate the $x$ variable:
1. $x - y - 2(-3) = 4$
2. $-x + 2y + (-3) = 1$
Multiply to simplify:
$x - y + 6 = 4$
$-x + 2y - 3 = 1$
Move constants to the right side of the equation:
$x - y = -2$
$-x + 2y = 4$
Add the equations together:
$y = 2$
Substitute these values for $y$ and $z$ into one of the original equations to solve for $x$. Use equation $1$:
1. $x - 2 - 2(-3) = 4$
Multiply to simplify:
$x - 2 + 6 = 4$
Combine like terms:
$x + 4 = 4$
Subtract $4$ from each side of the equation:
$x = 0$
The solution is $(0, 2, -3)$.
