Answer
$(1, -1, 2)$
Work Step by Step
Label the original equations:
1. $x + y + z = 2$
2. $x + 2z = 5$
3. $2x + y - z = -1$
Solve equation $2$ for $x$. This will be equation $4$:
4. $x = -2z + 5$
Substitute the expression for $x$ given by equation $4$ into equation $1$:
$(-2z + 5) + y + z = 2$
Combine like terms. This will be equation $5$:
5. $y - z + 5 = 2$
Move constants to the right side of the equation by subtracting $5$ from both sides of the equation:
5. $y - z = -3$
Substitute the expression for $x$ given by equation $4$ into equation $3$:
$2(-2z + 5) + y - z = -1$
Distribute and multiply to simplify. This will be equation $6$:
6. $-4z + 10 + y - z = -1$
Combine like terms on the left side of the equation:
6. $y - 5z + 10 = -1$
Move constants to the right side of the equation by subtracting $10$ from both sides of the equation:
6. $y - 5z = -11$
Set up a system of equations made up of equations $5$ and $6$:
5. $y - z = -3$
6. $y - 5z = -11$
Multiply equation $5$ by $-1$ so that the $y$ variable in the two equations differing in sign:
5. $-y + z = 3$
6. $y - 5z = -11$
Add the two equations together:
$-4z = -8$
Divide each side of the equation by $-4$ to solve for $z$:
$z = 2$
Substitute this value for $z$ into equation $4$ to find the value of $x$:
$x = -2(2) + 5$
Multiply to simplify:
$x = -4 + 5$
Add to solve for $x$:
$x = 1$
Substitute the values for $x$ and $z$ into equation $1$ to find the value of $y$:
$1 + y + 2 = 2$
Combine like terms on the left side of the equation:
$y + 3 = 2$
Subtract $3$ from both sides of the equation to move constants to the right side of the equation:
$y = -1$
The solution is $(1, -1, 2)$.
To check the solution, plug in the three values into one of the original equations. Use equation $3$:
$2(1) + (-1) - 2 = -1$
$2 - 1 - 2 = -1$
$-1 = -1$
The sides are equal to one another; therefore, the solution is correct.
