Solve {l}{x-2y=z}{3x+y-z=0}{x+5y-2z=4}

$$\left\{ \begin{array} { l } { x - 2 y = z } \\ { 3 x + y - z = 0 } \\ { x + 5 y - 2 z = 4 } \end{array} \right.$$

Solve for x, y, z

$$x=-\frac{4}{7}\approx -0.571428571$$

$$y=\frac{8}{21}\approx 0.380952381$$

$$z = -\frac{4}{3} = -1\frac{1}{3} \approx -1.333333333$$

Short Steps Using Substitution

Solve x-2y=z for z.

$$z=x-2y$$

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

$$3x+y-\left(x-2y\right)=0 x+5y-2\left(x-2y\right)=4$$

Solve these equations for y and x respectively.

$$y=-\frac{2}{3}x x=9y-4$$

Substitute -\frac{2}{3}x for y in the equation x=9y-4.

$$x=9\left(-\frac{2}{3}\right)x-4$$

Solve x=9\left(-\frac{2}{3}\right)x-4 for x.

$$x=-\frac{4}{7}$$

Substitute -\frac{4}{7} for x in the equation y=-\frac{2}{3}x.

$$y=-\frac{2}{3}\left(-\frac{4}{7}\right)$$

Calculate y from y=-\frac{2}{3}\left(-\frac{4}{7}\right).

$$y=\frac{8}{21}$$

Substitute \frac{8}{21} for y and -\frac{4}{7} for x in the equation z=x-2y.

$$z=-\frac{4}{7}-2\times \frac{8}{21}$$

Calculate z from z=-\frac{4}{7}-2\times \frac{8}{21}.

$$z=-\frac{4}{3}$$

The system is now solved.

$$x=-\frac{4}{7} y=\frac{8}{21} z=-\frac{4}{3}$$