Answer
The solution is $(-2, 1, 5)$.
Work Step by Step
Label the original equations:
1. $-2x + 9y - z = 8$
2. $3x - 4y + z = -5$
3. $5x + 5y - z = -10$
The first step is to choose two equations to work with where one variable can be eliminated. Let's choose equations $1$ and $2$. Modify the equations by multiplying them by a non-zero factor so that one variable is the same in both equations but differing in sign so that this variable can be eliminated when the two equations are added together.
In this case, no modification is necessary:
1. $-2x + 9y - z = 8$
2. $3x - 4y + z = -5$
Add the equations. This will become equation $4$:
4. $x + 5y = 3$
Now, choose another two equations again and modify them. Modify these equations such that the $z$ variable can be eliminated. This modified equation will be equation $5$ and will be added to equation $4$ to try to eliminate another variable.
Use equations $2$ and $3$. Again, no modification is necessary:
2. $3x - 4y + z = -5$
3. $5x + 5y - z = -10$
Add the equations together. This will become equation $5$:
5. $8x + y = -15$
Set up a system of equations consisting of equations $4$ and $5$:
4. $x + 5y = 3$
5. $8x + y = -15$
Modify these equations such that the $y$ variable can be eliminated. Multiply equation $5$ by $-5$ and leave equation $4$ as-is:
4. $x + 5y = 3$
5. $-5(8x + y) = -5(-15)$
Distribute and multiply to simplify:
4. $x + 5y = 3$
5. $-40x - 5y = 75$
Add the two equations together:
$-39x = 78$
Divide both sides by $-39$ to solve for $x$:
$x = -2$
Substitute this value for $x$ into equation $4$ to solve for $y$:
4. $-2 + 5y = 3$
Add $2$ to each side of the equation to move constants to the right side of the equation:
$5y = 5$
Divide both sides by $5$ to solve for $y$:
$y = 1$
Substitute the values for $x$ and $y$ into one of the original equations to find $z$. Use equation $2$:
2. $3(-2) - 4(1) + z = -5$
Multiply to simplify:
$-6 - 4 + z = -5$
Combine like terms on the left side of the equation:
$-10 + z = -5$
Add $10$ to each side of the equation to solve for $z$:
$z = 5$
The solution is $(-2, 1, 5)$.
Check the solution by plugging in the values into one of the original equations:
Use equation $1$:
1. $-2(-2) + 9(1) - 5 = 8$
Multiply to simplify:
1. $4 + 9 - 5 = 8$
Add or subtract from left to right:
$8 = 8$
Both sides are equal to one another; therefore, the solution is correct.
