Answer
$(4, 2, -3)$
Work Step by Step
Label the original equations first:
1. $x - y + z = -1$
2. $x + y + 3z = -3$
3. $2x - y + 2z = 0$
The first step is to combine equations $1$ and $2$ to eliminate the $y$ variable:
1. $x - y + z = -1$
2. $x + y + 3z = -3$
Add the equations together. Label this new equation $4$:
4. $2x + 4z = -4$
Combine equations $2$ and $3$ to eliminate the $y$ variable again:
2. $x + y + 3z = -3$
3. $2x - y + 2z = 0$
Add the two equations together. Label this new equation $5$:
5. $3x + 5z = -3$
Combine equations $4$ and $5$ to try to get rid of another variable:
4. $2x + 4z = -4$
5. $3x + 5z = -3$
Before adding the two equations, modify the equations so that one variable is the same in both equations but differs in sign. Multiply equation $4$ by $3$ and equation $5$ by $-2$. These will be equations $6$ and $7$:
6. $6x + 12z = -12$
7. $-6x - 10z = 6$
Add these two equations together:
$2z = -6$
Divide both sides by $2$:
$z = -3$
Substitute this value for $z$ into equation $4$ to solve for $x$:
$2x + 4(-3) = -4$
Multiply to simplify:
$2x - 12 = -4$
Add $12$ to both sides of the equation:
$2x = 8$
Divide each side by $2$:
$x = 4$
Substitute these values for $x$ and $z$ into one of the original equations to solve for $y$. Use equation $1$:
1. $4 - y + (-3) = -1$
Combine like terms:
$-y + 1 = -1$
Subtract $2$ from each side of the equation:
$-y = -2$
Divide both sides of the equation by $-1$ to solve for $y$:
$y = 2$
The solution is $(4, 2, -3)$.
