Answer
$(4, -1, 2)$
Work Step by Step
Label the original equations first:
1. $x - 2y + 3z = 12$
2. $2x - y - 2z = 5$
3. $2x + 2y - z = 4$
Combine equations $1$ and $3$ to eliminate the $y$ variable:
1. $x - 2y + 3z = 12$
3. $2x + 2y - z = 4$
Add the equations:
4. $3x + 2z = 16$
Combine equations $2$ and $3$ to eliminate the $y$ variable. Modify equation $2$ by multiplying it by $2$:
5. $2(2x - y - 2z) = 2(5)$
Use distributive property:
5. $4x - 2y - 4z = 10$
Combine equations $3$ and $5$:
3. $2x + 2y - z = 4$
5. $4x - 2y - 4z = 10$
Add the equations together:
6. $6x - 5z = 14$
Multiply equation $4$ by $-2$ to be able to add to equation $6$ to eliminate another variable:
7. $-2(3x + 2z) = -2(16)$
Use distributive property:
7. $-6x - 4z = -32$
Combine equations $6$ and $7$:
6. $6x - 5z = 14$
7. $-6x - 4z = -32$
Add the two equations:
$-9z = -18$
Divide each side of the equation by $-9$:
$z = 2$
Substitute this $z$ value into equation $4$ to find the value for $x$:
$3x + 2(2) = 16$
Multiply to simplify:
$3x + 4 = 16$
Subtract $4$ from both sides of the equation:
$3x = 12$
Divide both sides by $3$:
$x = 4$
Substitute the values for $x$ and $z$ into one of the original equations to find $y$. Let's use equation $3$:
3. $2(4) + 2y - 2 = 4$
Multiply to simplify:
$8 + 2y - 2 = 4$
Combine like terms on the left side of the equation:
$6 + 2y = 4$
Subtract $6$ from each side of the equation:
$2y = -2$
Divide both sides by $2$:
$y = -1$
The solution is $(4, -1, 2)$.
To check the solution, plug in the three values into one of the original equations. Use equation $2$:
$2(4) - (-1) - 2(2) = 5$
$8 + 1 - 4 = 5$
$5 = 5$
The sides are equal to one another; therefore, the solution is correct.
