Answer
$(1, -4, 3)$
Work Step by Step
Label the original equations first:
1. $3x + 3y + 6z = 9$
2. $2x + y + 3z = 7$
3. $x + 2y - z = -10$
Add equations $1$ and $2$ and modify them such that one variable is the same in both equations but differs only in sign. Multiply equation $2$ by $-3$:
4. $-3(2x + y + 3z) = -3(7)$
Use distributive property:
4. $-6x - 3y - 9z = -21$
Combine equations $1$ and $4$:
1. $3x + 3y + 6z = 9$
4. $-6x - 3y - 9z = -21$
Add the equations:
5. $-3x - 3z = -12$
Combine equations $2$ and $3$ to eliminate the $y$ variable. Modify equation $2$ by multiplying it by $-2$:
6. $-2(2x + y + 3z) = -2(7)$
Use distributive property:
6. $-4x - 2y - 6z = -14$
Combine equations $3$ and $6$:
3. $x + 2y - z = -10$
6. $-4x - 2y - 6z = -14$
Add the equations together:
7. $-3x - 7z = -24$
Combine equations $5$ and $7$, but first, one variable must be eliminated. Multiply equation $5$ by $-1$:
8. $-1(-3x - 3z) = -1(-12)$
Use distributive property:
8. $3x + 3z = 12$
Combine equations $7$ and $8$:
7. $-3x - 7z = -24$
8. $3x + 3z = 12$
Add the two equations:
$-4z = -12$
Divide each side of the equation by $-4$:
$z = 3$
Substitute this $z$ value into equation $8$ to find the value for $x$:
8. $3x + 3(3) = 12$
Multiply to simplify:
$3x + 9 = 12$
Subtract $9$ from both sides of the equation:
$3x = 3$
Divide both sides by $3$:
$x = 1$
Substitute the values for $x$ and $z$ into one of the original equations to find $y$. Let's use equation $3$:
3. $1 + 2y - 3 = -10$
Multiply to simplify:
$1 + 2y - 3 = -10$
Combine like terms on the left side of the equation:
$2y - 2 = -10$
Add $2$ to each side of the equation:
$2y = -8$
Divide both sides by $2$:
$y = -4$
The solution is $(1, -4, 3)$.
To check the solution, plug in the three values into one of the original equations. Use equation $2$:
$2(1) + (-4) + 3(3) = 7$
$2 - 4 + 9 = 7$
$7 = 7$
The sides are equal to one another; therefore, the solution is correct.