Answer
$(0, 3, 4)$
Work Step by Step
Label the original equations first:
1. $2x + 3y - 2z = 1$
2. $-x - y + 2z = 5$
3. $3x + 2y - 3z = -6$
The first step is to choose two equations to work with where one variable can be eliminated. Let's choose the second and third equations and modify them such that one variable is the same in both equations but differs only in sign.
Multiply the second equation by $2$. This will become the fourth equation:
4. $-2x - 2y + 4z = 10$
Combine the fourth equation and the third equation to eliminate the $x$ variable:
3. $3x + 2y - 3z = -6$
4. $-2x - 2y + 4z = 10$
Add the equations together:
5. $x + z = 4$
Modify the second equation again by multiplying it by $3$ such that the $y$ terms in both the first and second equations are the same but differ only in sign:
6. $-3x - 3y + 6z = 15$
Combine the first and sixth equations:
1. $2x + 3y - 2z = 1$
6. $-3x - 3y + 6z = 15$
Add the equations:
7. $-x + 4z = 16$
Combine the fifth and seventh equations:
5. $x + z = 4$
7. $-x + 4z = 16$
Add the two equations together to eliminate the $x$ variable:
$5z = 20$
Divide each side of the equation by $5$:
$z = 4$
Substitute this value for $z$ into the fifth equation to solve for $x$:
5. $x + 4 = 4$
Subtract $4$ from each side of the equation:
$x = 0$
Substitute the values for $x$ and $z$ into one of the original equations to find $y$. Let's use the second equation:
2. $-0 - y + 2(4) = 5$
Multiply to simplify:
$-0 - y + 8 = 5$
Combine like terms:
$-y + 8 = 5$
Subtract $8$ from each side of the equation:
$-y = -3$
Divide both sides of the equation by $-1$ to isolate $y$:
$y = 3$
The solution is $(0, 3, 4)$.
