Answer
$(0, 1, 7)$
Work Step by Step
Label the original equations first:
1. $x + 3y - z = -4$
2. $2x - y + 2z = 13$
3. $3x - 2y - z = -9$
Solve equation $3$ for $z$. This will become equation $4$:
4. $z = 3x - 2y + 9$
Substitute the expression for $z$ into equation $1$:
$x + 3y - (3x - 2y + 9) = -4$
Use distributive property to simplify:
$x + 3y - 3x + 2y - 9 = -4$
Combine like terms:
$-2x + 5y - 9 = -4$
Add $9$ to both sides of the equation. This will become equation $5$:
5. $-2x + 5y = 5$
Substitute the expression for $z$ into equation $2$:
$2x - y + 2(3x - 2y + 9) = 13$
Distribute to simplify:
$2x - y + 6x - 4y + 18 = 13$
Combine like terms:
$8x - 5y + 18 = 13$
Subtract $18$ from each side of the equation. This will become equation $6$:
6. $8x - 5y = -5$
Set up a system of equations made up of equations $5$ and $6$:
$-2x + 5y = 5$
$8x - 5y = -5$
Multiply equation $5$ by $4$:
$4(-2x + 5y) = 4(5)$
Distribute and multiply:
$-8x + 20y = 20$
Set up the system of equations:
5. $-8x + 20y = 20$
6. $8x - 5y = -5$
Add the equations:
$15y = 15$
Divide both sides of the equation by $15$ to solve for $y$:
$y = 1$
Substitute this value for $y$ into equation $6$ to find the value of $x$:
$8x - 5(1) = -5$
Multiply to simplify:
$8x - 5 = -5$
Add $5$ to each side of the equation:
$8x = 0$
Divide each side of the equation by $8$ to solve for $x$:
$x = 0$
Substitute the values for $x$ and $y$ into equation $1$ to find the value of $z$:
$0 + 3(1) - z = -4$
Multiply to simplify:
$0 + 3 - z = -4$
Add like terms on the left side of the equation:
$3 - z = -4$
Subtract $3$ from each side of the equation:
$-z = -7$
Divide each side by $-1$ to solve for $z$:
$z = 7$
The solution is $(0, 1, 7)$.
To check the solution, plug in the three values into one of the original equations. Use equation $3$:
$3(0) - 2(1) - 7 = -9$
$0 - 2 - 7 = -9$
$-9 = -9$
The sides are equal to one another; therefore, the solution is correct.