Answer
$(-3,-4,-5)$
Work Step by Step
$x+y-\frac{3}{2}z=\frac{1}{2}$ Equation $(1)$
$-y-2z=14 $ Equation $(2)$
$x-\frac{2}{3}y=-\frac{1}{3}$ Equation $(3)$
Multiply both sides of Equation $(1)$ by $2$ and Equation $(3)$ by $3$ to eliminate fractions, we get
$2x+2y-3z = 1$ Equation $(4)$
$-y-2z=14 $ Equation $(5)$
$3x-2y=-1$ Equation $(6)$
Adding Equation $(4)$ and Equation $(6)$
$2x+3x+2y-2y-3z = 1-1$
$5x-3z = 0$ Equation $(7)$
Multiply Equation $(5)$ by $-2$ then add with Equation $(6)$
$-2(-y-2z) + 3x-2y = -2(14)-1$
$2y+4z + 3x -2y = -28 -1$
$3x+4z = -29$ Equation $(8)$
Multiply Equation $(7)$ by $4$ and Equation $(8)$ by $3$ then add
$4(5x-3z)+ 3(3x+4z) = 0 +3(-29)$
$20x-12z+9x+12z = -87$
$29x = -87$
$x = -3$
Substituting $x$ value in Equation $(6)$
$3x-2y=-1$
$3(-3)-2y = -1$
$-9-2y = -1$
$-2y = 8$
$y= -4$
Substituting $y$ value in Equation $(5)$
$-y-2z=14 $
$-(-4)-2z = 14$
$4-2z = 14$
$-2z = 10$
$z = -5$
Solution: $(-3,-4,-5)$
