Answer
$(2, 1, -5)$
Work Step by Step
Label the original equations first:
1. $-2x + y - z = 2$
2. $-x - 3y + z = -10$
3. $3x + 6z = -24$
Divide equation $3$ by $3$ so that the terms are the simplest possible:
4. $x + 2z = -8$
Add the first and second equations and modify them such that one variable is the same in both equations but differs only in sign. Since equation $3$ is missing a $y$ term, modify the first and second equations so that the $y$ term can be eliminated as well. Multiply equation $1$ by $3$:
5. $3(-2x + y - z) = 3(2)$
Use distributive property:
5. $-6x + 3y - 3z = 6$
Combine equations $2$ and $5$:
2. $-x - 3y + z = -10$
5. $-6x + 3y - 3z = 6$
Add the equations:
6. $-7x - 2z = -4$
Combine equations $4$ and $6$ to eliminate a variable:
4. $x + 2z = -8$
6. $-7x - 2z = -4$
Add the two equations together:
$-6x = -12$
Divide both sides of the equation by $-6$:
$x = 2$
Substitute this $x$ value into equation $4$ to find the value for $z$:
4. $2 + 2z = -8$
Subtract $2$ from both sides of the equation:
$2z = -10$
Divide both sides by $2$:
$z = -5$
Substitute the values for $x$ and $z$ into one of the original equations to find $y$. Let's use the second equation:
2. $-2 - 3y + (-5) = -10$
Combine like terms on the left side of the equation:
$-7 - 3y = -10$
Add $7$ to each side of the equation:
$-3y = -3$
Divide both sides of the equation by $-3$ to solve for $y$:
$y = 1$
The solution is $(2, 1, -5)$.
To check the solution, plug in the three values into one of the original equations. Use equation $1$:
$-2(2) + 1 - (-5) = 2$
$-4 + 1 + 5 = 2$
$2 = 2$
The sides are equal to one another; therefore, the solution is correct.