Solve {l}{2x+y+z=0}{x-y=1}{z-2x=pi}

$$\left\{ \begin{array} { l } { 2 x + y + z = 0 } \\ { x - y = 1 } \\ { z - 2 x = \pi } \end{array} \right.$$

Solve for x, y, z

$$x=\frac{1-\pi }{5}\approx -0.428318531$$

$$y=\frac{-\pi -4}{5}\approx -1.428318531$$

$$z = \frac{3 \pi + 2}{5} \approx 2.284955592$$

Short Steps Using Substitution

Solve 2x+y+z=0 for y.

$$y=-2x-z$$

Substitute -2x-z for y in the equation x-y=1.

$$x-\left(-2x-z\right)=1$$

Solve the second equation for x and the third equation for z.

$$x=-\frac{1}{3}z+\frac{1}{3} z=2x+\pi$$

Substitute -\frac{1}{3}z+\frac{1}{3} for x in the equation z=2x+\pi .

$$z=2\left(-\frac{1}{3}z+\frac{1}{3}\right)+\pi$$

Solve z=2\left(-\frac{1}{3}z+\frac{1}{3}\right)+\pi for z.

$$z=\frac{3}{5}\pi +\frac{2}{5}$$

Substitute \frac{3}{5}\pi +\frac{2}{5} for z in the equation x=-\frac{1}{3}z+\frac{1}{3}.

$$x=-\frac{1}{3}\left(\frac{3}{5}\pi +\frac{2}{5}\right)+\frac{1}{3}$$

Calculate x from x=-\frac{1}{3}\left(\frac{3}{5}\pi +\frac{2}{5}\right)+\frac{1}{3}.

$$x=\frac{1}{5}-\frac{1}{5}\pi$$

Substitute \frac{1}{5}-\frac{1}{5}\pi for x and \frac{3}{5}\pi +\frac{2}{5} for z in the equation y=-2x-z.

$$y=-2\left(\frac{1}{5}-\frac{1}{5}\pi \right)-\left(\frac{3}{5}\pi +\frac{2}{5}\right)$$

Calculate y from y=-2\left(\frac{1}{5}-\frac{1}{5}\pi \right)-\left(\frac{3}{5}\pi +\frac{2}{5}\right).

$$y=-\frac{4}{5}-\frac{1}{5}\pi$$

The system is now solved.

$$x=\frac{1}{5}-\frac{1}{5}\pi y=-\frac{4}{5}-\frac{1}{5}\pi z=\frac{3}{5}\pi +\frac{2}{5}$$