Answer
$(5, -2, 0)$
Work Step by Step
Label the original equations:
1. $x - y + 2z = 7$
2. $2x + y + z = 8$
3. $x - z = 5$
Solve equation $3$ for $x$. This will be equation $4$:
4. $x = z + 5$
Substitute the expression for $x$ given by equation $4$ into equation $1$:
$(z + 5) - y + 2z = 7$
Combine like terms. This will be equation $5$:
5. $-y + 3z + 5 = 7$
Move constants to the right side of the equation by subtracting $5$ from both sides of the equation:
5. $-y + 3z = 2$
Substitute the expression for $x$ given by equation $4$ into equation $2$:
$2(z + 5) + y + z = 8$
Distribute and multiply to simplify. This will be equation $6$:
6. $2z + 10 + y + z = 8$
Combine like terms on the left side of the equation:
6. $y + 3z + 10 = 8$
Move constants to the right side of the equation by subtracting $10$ from both sides of the equation:
6. $y + 3z = -2$
Set up a system of equations made up of equations $5$ and $6$:
$-y + 3z = 2$
$y + 3z = -2$
No modification is needed here because the $y$ variable in the two equations are the same, differing only in sign.
Add the two equations together:
$6z = 0$
Divide each side of the equation by $6$ to solve for $z$:
$z = 0$
Substitute this value for $z$ into equation $4$ to find the value of $x$:
$x = 0 + 5$
Add to solve for $x$:
$x = 5$
Substitute the values for $x$ and $z$ into equation $1$ to find the value of $y$:
$5 - y + 2(0) = 7$
Combine like terms on the left side of the equation:
$-y + 5 = 7$
Subtract $5$ from both sides of the equation to move constants to the right side of the equation:
$-y = 2$
Divide both sides of the equation by $-1$:
$y = -2$
The solution is $(5, -2, 0)$.
To check the solution, plug in the three values into one of the original equations. Use equation $2$:
$2(5) + (-2) + 0 = 8$
$10 - 2 + 0 = 8$
$8 = 8$
The sides are equal to one another; therefore, the solution is correct.