Answer
$(5, 2, 2)$
Work Step by Step
Label the original equations first:
1. $13 = 3x - y$
2. $4y - 3x + 2z = -3$
3. $z = 2x - 4y$
Substitute the expression for $z$ given by equation $3$ into equation $2$:
$4y - 3x + 2(2x - 4y) = -3$
Use distributive property to simplify:
$4y - 3x + 4x - 8y = -3$
Combine like terms:
$x - 4y = -3$
Transform equation $4$ so that we can add it to equation $1$. To do this, multiply equation $4$ by $-3$:
$-3(x - 4y) = -3(-3)$
Distribute and multiply to simplify:
$-3x + 12y = 9$
Set up a system of equations made up of equations $1$ and $5$:
$3x - y = 13$
$-3x + 12y = 9$
Add the two equations together:
$11y = 22$
Divide each side of the equation by $11$:
$y = 2$
Substitute this value for $y$ into equation $1$ to find the value of $x$:
$13 = 3x - 2$
Add $2$ to each side of the equation to move constants to one side of the equation:
$3x = 15$
Divide each side by $3$ to solve for $x$:
$x = 5$
Substitute the values for $x$ and $y$ into equation $3$ to find the value of $z$:
$z = 2(5) - 4(2)$
Multiply to simplify:
$z = 10 - 8$
Subtract to solve for $z$:
$z = 2$
The solution is $(5, 2, 2)$.
To check the solution, plug in the three values into one of the original equations. Use equation $2$:
$4(2) - 3(5) + 2(2) = -3$
$8 - 15 + 4 = -3$
$-3 = -3$
The sides are equal to one another; therefore, the solution is correct.
