Answer
$(12, 6, 4)$
Work Step by Step
$\frac{3}{4}x-\frac{1}{3}y+\frac{1}{2}z=9$ Equation $(1)$
$\frac{1}{6}x+\frac{1}{3}y-\frac{1}{2}z=2$ Equation $(2)$
$\frac{1}{2}x-y+\frac{1}{2}z=2$ Equation $(3)$
Equation $(1)$ + Equation $(2)$
$(\frac{3}{4}x-\frac{1}{3}y+\frac{1}{2}z) + (\frac{1}{6}x+\frac{1}{3}y-\frac{1}{2}z) = 9+2$
$\frac{11}{12}x=11$
$x=12$
Equation $(2)$ + Equation $(3)$
$(\frac{1}{6}x+\frac{1}{3}y-\frac{1}{2}z) + (\frac{1}{2}x-y+\frac{1}{2}z)=2+2$
$\frac{4}{6}x-\frac{2}{3}y=4$ Equation $(4)$
Substitute $x=12$ into Equation $(4)$ to get $y$.
$(\frac{4}{6}\times12)-\frac{2}{3}y=4$
$8-\frac{2}{3}y=4$
$-\frac{2}{3}y=-4$
$y=6$
Substitute known values for $x$ and $y$ into Equation $(3)$ to get $z$.
$(\frac{1}{2}\times12) - 6 + \frac{1}{2}z=2$
$6-6+\frac{1}{2}z=2$
$\frac{1}{2}z=2$
$z=4$
$(12, 6, 4)$ satisfies all of the given equations.
Solution is $(12, 6, 4)$.