Solve {l}{x+3/2-y-1/2+z+2/4=3/2}{frac{x-5}{2}+frac{y+1}{3}-frac{z}{4}=-frac{25}{6}}{frac{x-3}{4}-frac{y+1}{2}+frac{z-3}{2}=-frac{5}{2}}

$$\left. \begin{array} { l } { \frac { x + 3 } { 2 } - \frac { y - 1 } { 2 } + \frac { z + 2 } { 4 } = \frac { 3 } { 2 } } \\ { \frac { x - 5 } { 2 } + \frac { y + 1 } { 3 } - \frac { z } { 4 } = - \frac { 25 } { 6 } } \\ { \frac { x - 3 } { 4 } - \frac { y + 1 } { 2 } + \frac { z - 3 } { 2 } = - \frac { 5 } { 2 } } \end{array} \right.$$

Solve for x, y, z

$$x=-3$$

$$y=0$$

$$z=2$$

Short Steps Using Substitution

Multiply each equation by the least common multiple of denominators in it. Simplify.

$$10+2x-2y+z=6 -26+6x+4y-3z=-50 -11+x-2y+2z=-10$$

Solve 10+2x-2y+z=6 for z.

$$z=-4-2x+2y$$

Substitute -4-2x+2y for z in the second and third equation.

$$-26+6x+4y-3\left(-4-2x+2y\right)=-50 -11+x-2y+2\left(-4-2x+2y\right)=-10$$

Solve these equations for y and x respectively.

$$y=18+6x x=-3+\frac{2}{3}y$$

Substitute 18+6x for y in the equation x=-3+\frac{2}{3}y.

$$x=-3+\frac{2}{3}\left(18+6x\right)$$

Solve x=-3+\frac{2}{3}\left(18+6x\right) for x.

$$x=-3$$

Substitute -3 for x in the equation y=18+6x.

$$y=18+6\left(-3\right)$$

Calculate y from y=18+6\left(-3\right).

$$y=0$$

Substitute 0 for y and -3 for x in the equation z=-4-2x+2y.

$$z=-4-2\left(-3\right)+2\times 0$$

Calculate z from z=-4-2\left(-3\right)+2\times 0.

$$z=2$$

The system is now solved.

$$x=-3 y=0 z=2$$