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 choose two equations to work with where one variable can be eliminated. Let's choose the first and second equations. In this case, no modification is necessary because the $y$ variable is already the same in both equations, just differing in sign, so when the two equations are added, the $y$ variable can be eliminated:
1. $x - y + z = -1$
2. $x + y + 3z = -3$
Add the equations. This will become equation $4$:
4. $2x + 4z = -4$
Now, choose another two equation again and modify them. Modify these equations such that the $y$ variable can be eliminated. This modified equation will be equation $5$ and will be added to equation $4$ to try to eliminate another variable.
Choose equations $2$ and $3$. This time, no modification is necessary:
2. $x + y + 3z = -3$
3. $2x - y + 2z = 0$
Add the equations. This will become equation $5$:
5. $3x + 5z = -3$
Modify equations $4$ and $5$ such that one variable is the same but differing in sign. Multiply equation $4$ by $-3$ and equation $5$ by $2$:
4. $-3(2x + 4z) = -3(-4)$
5. $2(3x + 5z) = 2(-3)$
Distribute and multiply to simplify:
$-6x - 12z = 12$
$6x + 10z = -6$
Add the equations together:
$-2z = 6$
Divide both sides by $-2$ to solve for $z$:
$z = -3$
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 each side of the equation to move constants to the right side of the equation:
$2x = 8$
Divide both sides by $2$ to solve for $x$:
$x = 4$
Substitute the values for $x$ and $z$ into one of the original equations to find $y$. Use equation $1$:
$4 - y + (-3) = -1$
$-y + 1 = -1$
Subtract $1$ from each side of the equation:
$-y = -2$
Divide both sides of the equation by $-1$ to isolate $y$:
$y = 2$
The solution is $(4, 2, -3)$.
Check the solution by plugging in the values into each of the equations:
Equation $1$:
$4 - 2 + (-3) = -1$
$-1 = -1$
Equation $2$:
$4 + 2 + 3(-3) = -3$
$4 + 2 - 9 = -3$
$-3 = -3$
Equation $3$:
$2(4) - 2 + 2(-3) = 0$
$8 - 2 - 6 = 0$
$0 = 0$
When plugging the solution into each of the original equations, both sides are equal to one another; therefore, the solution is correct.
