Solve {l}{x-30/10+x-y/10=0}{y-x/10+y/10+frac{y-z}{20}+frac{y-z+20}{20}=0}{frac{z-y}{10}+frac{z}{10}+frac{z-y-20}{20}=0}

$$\left. \begin{array} { l } { \frac { x - 30 } { 10 } + \frac { x - y } { 10 } = 0 } \\ { \frac { y - x } { 10 } + \frac { y } { 10 } + \frac { y - z } { 20 } + \frac { y - z + 20 } { 20 } = 0 } \\ { \frac { z - y } { 10 } + \frac { z } { 10 } + \frac { z - y - 20 } { 20 } = 0 } \end{array} \right.$$

Solve for x, y, z

$$x = \frac{330}{19} = 17\frac{7}{19} \approx 17.368421053$$

$$y = \frac{90}{19} = 4\frac{14}{19} \approx 4.736842105$$

$$z = \frac{130}{19} = 6\frac{16}{19} \approx 6.842105263$$

Short Steps Using Substitution

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

$$-30+2x-y=0 20-2x+6y-2z=0 -20+5z-3y=0$$

Solve -30+2x-y=0 for y.

$$y=-30+2x$$

Substitute -30+2x for y in the second and third equation.

$$20-2x+6\left(-30+2x\right)-2z=0 -20+5z-3\left(-30+2x\right)=0$$

Solve these equations for x and z respectively.

$$x=16+\frac{1}{5}z z=-14+\frac{6}{5}x$$

Substitute 16+\frac{1}{5}z for x in the equation z=-14+\frac{6}{5}x.

$$z=-14+\frac{6}{5}\left(16+\frac{1}{5}z\right)$$

Solve z=-14+\frac{6}{5}\left(16+\frac{1}{5}z\right) for z.

$$z=\frac{130}{19}$$

Substitute \frac{130}{19} for z in the equation x=16+\frac{1}{5}z.

$$x=16+\frac{1}{5}\times \frac{130}{19}$$

Calculate x from x=16+\frac{1}{5}\times \frac{130}{19}.

$$x=\frac{330}{19}$$

Substitute \frac{330}{19} for x in the equation y=-30+2x.

$$y=-30+2\times \frac{330}{19}$$

Calculate y from y=-30+2\times \frac{330}{19}.

$$y=\frac{90}{19}$$

The system is now solved.

$$x=\frac{330}{19} y=\frac{90}{19} z=\frac{130}{19}$$