Answer
$(0, 3, -2)$
Work Step by Step
Label the original equations first:
1. $x - y + 2z = -7$
2. $y + z = 1$
3. $x = 2y + 3z$
Rewrite equation $3$ so that all variables are on the left side of the equation:
4. $x - 2y - 3z = 0$
Modify one of the equations so that the $x$ term can be eliminated. This will allow us to be able to add this equation to equation $2$ to eliminate one variable and be able to solve for the other. Multiply equation $1$ by $-1$:
5. $-1(x - y + 2z) = -1(-7)$
Use distributive property:
5. $-x + y - 2z = 7$
Combine equations $4$ and $5$:
4. $x - 2y - 3z = 0$
5. $-x + y - 2z = 7$
Add the equations together:
7. $-y - 5z = 7$
Combine equations $2$ and $7$:
2. $y + z = 1$
7. $-y - 5z = 7$
Add the equations:
$-4z = 8$
Divide both sides of the equation by $-4$:
$z = -2$
Substitute this value into equation $2$ to solve for $y$:
2. $y + (-2) = 1$
Add $2$ to each side of the equation:
$y = 3$
Substitute the values of $y$ and $z$ into one of the original equations. Let's use equation $3$:
3. $x = 2(3) + 3(-2)$
Multiply to simplify:
$x = 6 - 6$
Combine like terms to solve for $x$:
$x = 0$
The solution is $(0, 3, -2)$.
To check the solution, plug in the three values into one of the original equations. Use equation $1$:
$0 - 3 + 2(-2) = -7$
$0 - 3 - 4 = -7$
$-7 = -7$
The sides are equal to one another; therefore, the solution is correct.