Answer
$\{(x,y,z)\}=\{(4,8,6)\}$.
Work Step by Step
The given system of equations is
$\left\{\begin{matrix}
\frac{x+2}{6}& &-&\frac{y+4}{3}&+&\frac{z}{2}&=&0\\
\frac{x+1}{2}& &+&\frac{y-1}{2}&-&\frac{z}{4}&=&\frac{9}{2}\\
\frac{x-5}{4}& &+&\frac{y+1}{3}&+&\frac{z-2}{2}&=&\frac{19}{4}
\end{matrix}\right.$
First equation:-
$\Rightarrow \frac{x+2}{6}-\frac{y+4}{3}+\frac{z}{2}=0$
Multiply both sides by $6$.
$\Rightarrow 6\cdot \left (\frac{x+2}{6}-\frac{y+4}{3}+\frac{z}{2}\right )=6\cdot 0$
Apply distributive property.
$\Rightarrow x+2-2y-8+3z= 0$
$\Rightarrow x-2y-6+3z= 0$
Add $6$ to both sides.
$\Rightarrow x-2y-6+3z+6= 0+6$
Simplify.
$\Rightarrow x-2y+3z= 6$...... (1)
Second equation:-
$\Rightarrow \frac{x+1}{2}+\frac{y-1}{2}-\frac{z}{4}=\frac{9}{2}$
Multiply both sides by $4$.
$\Rightarrow 4\cdot \left (\frac{x+1}{2}+\frac{y-1}{2}-\frac{z}{4}\right )=4\cdot \frac{9}{2}$
Apply distributive property.
$\Rightarrow 2x+2+2y-2-z= 18$
$\Rightarrow 2x+2y-z= 18$...... (2)
Third equation:-
$\Rightarrow \frac{x-5}{4}+\frac{y+1}{3}+\frac{z-2}{2}=\frac{19}{4}$
Multiply both sides by $12$.
$\Rightarrow 12\cdot \left (\frac{x-5}{4}+\frac{y+1}{3}+\frac{z-2}{2}\right )=12\cdot \frac{19}{4}$
Apply distributive property.
$\Rightarrow 3x-15+4y+4+6z-12= 57$
$\Rightarrow 3x+4y+6z-23= 57$
Add $23$ to both sides
$\Rightarrow 3x+4y+6z-23+23= 57+23$
Simplify.
$\Rightarrow 3x+4y+6z= 80$...... (3)
Multiply equation (2) by $3$ and add to equation (1).
$\Rightarrow 3(2x+2y-z)+x-2y+3z= 3(18)+6$
Apply distributive property.
$\Rightarrow 6x+6y-3z+x-2y+3z= 54+6$
Add like terms.
$\Rightarrow 7x+4y= 60$ ...... (4)
Multiply equation (2) by $6$ and add to equation (3).
$\Rightarrow 6(2x+2y-z)+3x+4y+6z= 6(18)+80$
Apply distributive property.
$\Rightarrow 12x+12y-6z+3x+4y+6z=108+ 80$
Add like terms.
$\Rightarrow 15x+16y= 188$ ...... (5)
Multiply equation (4) by $-4$ and add to equation (5)
$\Rightarrow -4(7x+4y)+15x+16y=-4( 60)+188$
Apply distributive property.
$\Rightarrow -28x-16y+15x+16y=-240+188$
Add like terms.
$\Rightarrow -13x=-52$
Divide both sides $-13$.
$\Rightarrow \frac{-13x}{-13}=\frac{-52}{-13}$
Simplify.
$\Rightarrow x=4$
Substitute the value of $x$ into equation (4).
$\Rightarrow 7(4)+4y= 60$
Simplify.
$\Rightarrow 28+4y= 60$
Subtract $28$ from both sides.
$\Rightarrow 28+4y-28= 60-28$
Simplify.
$\Rightarrow 4y= 32$
Divide both sides by $4$.
$\Rightarrow \frac{4y}{4}= \frac{32}{4}$
Simplify.
$\Rightarrow y=8$
Substitute the values of $x$ and $y$ into equation (1).
$\Rightarrow 4-2(8)+3z= 6$
Simplify.
$\Rightarrow 4-16+3z= 6$
$\Rightarrow -12+3z= 6$
Add $12$ to both sides.
$\Rightarrow -12+3z+12= 6+12$
Simplify.
$\Rightarrow 3z= 18$
Divide both sides $3$.
$\Rightarrow \frac{3z}{3}= \frac{18}{3}$
Simplify.
$\Rightarrow z= 6$
The solution set is $\{(x,y,z)\}=\{(4,8,6)\}$.
