Answer
The solution is $(7, 1, -1)$.
Work Step by Step
Label the original equations:
1. $x - 9y + 8z = -10$
2. $x + y - z = 9$
3. $-x - 9z = 2$
The first step is to choose two equations to work with where one variable can be eliminated. Since equation $3$ is missing the $y$ variable, modify equations $1$ and $2$ such that when they are added together, the $y$ variable can be eliminated.
Multiply equation $2$ by $9$ and leave equation $1$ as-is:
1. $x - 9y + 8z = -10$
2. $9(x + y - z) = 9(9)$
Use distributive property and then multiply:
1. $x - 9y + 8z = -10$
2. $9x + 9y - 9z = 81$
Add the equations. This will become equation $4$:
4. $10x - z = 71$
Set up a system of equations made up of equations $3$ and $4$:
3. $-x - 9z = 2$
4. $10x - z = 71$
Modify these equations such that the $x$ variable can be eliminated. Multiply equation $3$ by $10$ and leave equation $4$ as-is:
3. $10(-x - 9z) = 10(2)$
4. $10x - z = 71$
Use distributive property and then multiply:
3. $-10x - 90z = 20$
4. $10x - z = 71$
Add the equations together:
$-91z = 91$
Divide both sides of the equation by $-91$ to solve for $z$:
$z = -1$
Substitute this value for $z$ into equation $3$ to solve for $x$:
3. $-x - 9(-1) = 2$
Add $2$ to each side of the equation to move constants to the right side of the equation:
$-x + 9 = 2$
Subtract $9$ from each side of the equation:
$-x = -7$
Divide both sides by $-1$ to solve for $x$:
$x = 7$
Substitute the values for $x$ and $z$ into one of the original equations to find $y$. Use equation $2$:
2. $7 + y - (-1) = 9$
Multiply to simplify:
2. $7 + y + 1 = 9$
Combine like terms on the left side of the equation:
$8 + y = 9$
Subtract $8$ from each side of the equation to solve for $y$:
$y = 1$
The solution is $(7, 1, -1)$.
Check the solution by plugging in the values into one of the original equations:
Use equation $1$:
1. $7 - 9(1) + 8(-1) = -10$
Multiply to simplify:
$7 - 9 - 8 = -10$
Add or subtract from left to right:
$-10 = -10$
Both sides are equal to one another; therefore, the solution is correct.
