Answer
$(2, 1, -5)$
Work Step by Step
Label the original equations first:
1. $2x - y + z = -2$
2. $x + 3y - z = 10$
3. $x + 2z = -8$
The first step is to isolate the $x$ variable in equation $3$. Subtract $2z$ from both sides of the equation:
$x = -2z - 8$
Substitute this expression for $x$ into both equations $1$ and $2$:
4. $2(-2z - 8) - y + z = -2$
5. $(-2z - 8) + 3y - z = 10$
Use distributive property:
4. $-4z - 16 - y + z = -2$
5. $-2z - 8 + 3y - z = 10$
Combine like terms:
4. $-y - 3z = 14$
5. $3y - 3z = 18$
Modify equation $4$ such that one variable is the same in equations $4$ and $5$ but differs in sign. Multiply equation $4$ by $-1$:
6. $y + 3z = -14$
Combine equations $5$ and $6$ to eliminate the $z$ variable:
5. $3y - 3z = 18$
6. $y + 3z = -14$
Add the equations together:
5. $4y = 4$
Divide both sides by $4$ to solve for $y$:
$y = 1$
Substitute this value for $y$ into the sixth equation to solve for $z$:
6. $1 + 3z = -14$
Subtract $1$ from each side of the equation:
$3z = -15$
Divide both sides of the equation by $3$ to solve for $z$:
$z = -5$
Substitute the values for $y$ and $z$ we just found into one of the original equations to solve for $x$. Use the first equation:
1. $2x - (1) + (-5) = -2$
Combine like terms on the left side of the equation:
$2x - 6 = -2$
Add $6$ to each side of the equation:
$2x = 4$
Divide each side of the equation by $2$:
$x = 2$
The solution is $(2, 1, -5)$.
