Answer
$(5, -3, -2)$
Work Step by Step
Label the original equations first:
1. $2y - 3z = 0$
2. $x + 3y = -4$
3. $3x + 4y = 3$
The first step is to choose two equations to work with where one variable can be eliminated. Let's choose the second and third equations and modify them such that one variable is the same in both equations but differ only in sign.
Multiply the second equation by $-3$. This will become the fourth equation:
4. $-3x - 9y = 12$
Combine the fourth equation and the third equation to eliminate the $x$ variable:
3. $3x + 4y = 3$
4. $-3x - 9y = 12$
Add the equations together:
$-5y = 15$
Divide both sides of the equation by $-5$:
$y = -3$
Substitute this $y$ value into the first equation to find the value of $z$:
1. $2(-3) - 3z = 0$
Multiply to simplify:
$-6 - 3z = 0$
Add $6$ to each side of the equation:
$-3z = 6$
Divide both sides of the equation by $-3$:
$z = -2$
Substitute the $y$ value into the second equation to find $x$:
2. $x + 3(-3) = -4$
Multiply to simplify:
$x - 9 = -4$
Add $9$ to each side of the equation:
$x = 5$
The solution is $(5, -3, -2)$.
