Answer
$\{\left ( -1,2,-2\right )\}$.
Work Step by Step
The given system of equations is
$\Rightarrow 2x-y+2z= -8$...... (1)
$\Rightarrow x+2y-3z= 9 $...... (2)
$\Rightarrow 3x-y-4z= 3$...... (3)
Multiply equation (1) by $2$ and add to equation (2).
$\Rightarrow 2(2x-y+2z)+x+2y-3z= 2(-8)+9$
Apply distributive property.
$\Rightarrow 4x-2y+4z+x+2y-3z= -16+9$
Simplify.
$\Rightarrow 5x+z= -7$...... (4).
Multiply equation (3) by $2$ and add to equation (2).
$\Rightarrow 2(3x-y-4z)+x+2y-3z= 2(3)+9$
Apply distributive property.
$\Rightarrow 6x-2y-8z+x+2y-3z= 6+9$
Simplify.
$\Rightarrow 7x-11z= 15$...... (5).
Multiply equation (4) by $11$ and add to equation (5).
$\Rightarrow 11(5x+z)+7x-11z= 11(-7)+15$
Apply distributive property.
$\Rightarrow 55x+11z+7x-11z= -77+15$
$\Rightarrow 62x= -62$
Divide both sides by $62$.
$\Rightarrow \frac{62x}{62}= \frac{-62}{62}$
Simplify.
$\Rightarrow x= -1$
Plug the value of $x$ into equation (4).
$\Rightarrow 5(-1)+z= -7$
Simplify.
$\Rightarrow -5+z= -7$
Add $5$ to both sides.
$\Rightarrow -5+z+5= -7+5$
Simplify.
$\Rightarrow z= -2$
Substitute the values of $x$ and $z$ into equation (2).
$\Rightarrow -1+2y-3(-2)= 9 $
Simplify.
$\Rightarrow -1+2y+6= 9 $
$\Rightarrow 2y+5= 9 $
Subtract $5$ from both sides.
$\Rightarrow 2y+5-5= 9-5 $
Add like terms.
$\Rightarrow 2y= 4 $
Divide both sides by $2$.
$\Rightarrow \frac{2y}{2}= \frac{4}{2} $
Simplify.
$\Rightarrow y= 2 $
The solution set is $\{\left ( x,y,z\right )\}=\{\left ( -1,2,-2\right )\}$.
