Solve {l}{x+y/7=x+4/5}{x-z/5=y-4/2}{frac{y-z}{3}=frac{x+2}{16}}

$$\left. \begin{array} { l } { \frac { x + y } { 7 } = \frac { x + 4 } { 5 } } \\ { \frac { x - z } { 5 } = \frac { y - 4 } { 2 } } \\ { \frac { y - z } { 3 } = \frac { x + 2 } { 16 } } \end{array} \right.$$

Solve for x, y, z

$$x = -\frac{738}{17} = -43\frac{7}{17} \approx -43.411764706$$

$$y = -\frac{200}{17} = -11\frac{13}{17} \approx -11.764705882$$

$$z=-4$$

Short Steps Using Substitution

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

$$5x+5y=7x+28 2x-2z=5y-20 16y-16z=3x+6$$

Solve 5x+5y=7x+28 for x.

$$x=\frac{5}{2}y-14$$

Substitute \frac{5}{2}y-14 for x in the second and third equation.

$$2\left(\frac{5}{2}y-14\right)-2z=5y-20 16y-16z=3\left(\frac{5}{2}y-14\right)+6$$

Solve these equations for z and y respectively.

$$z=-4 y=-\frac{72}{17}+\frac{32}{17}z$$

Substitute -4 for z in the equation y=-\frac{72}{17}+\frac{32}{17}z.

$$y=-\frac{72}{17}+\frac{32}{17}\left(-4\right)$$

Calculate y from y=-\frac{72}{17}+\frac{32}{17}\left(-4\right).

$$y=-\frac{200}{17}$$

Substitute -\frac{200}{17} for y in the equation x=\frac{5}{2}y-14.

$$x=\frac{5}{2}\left(-\frac{200}{17}\right)-14$$

Calculate x from x=\frac{5}{2}\left(-\frac{200}{17}\right)-14.

$$x=-\frac{738}{17}$$

The system is now solved.

$$x=-\frac{738}{17} y=-\frac{200}{17} z=-4$$