Solve {l}{2x+y+z=0}{x-y-z=4}{x+2y-4z=10}

$$\left. \begin{array} { l } { 2 x + y + z = 0 } \\ { x - y - z = 4 } \\ { x + 2 y - 4 z = 10 } \end{array} \right.$$

Solve for x, y, z

$$x = \frac{4}{3} = 1\frac{1}{3} \approx 1.333333333$$

$$y=-\frac{1}{3}\approx -0.333333333$$

$$z = -\frac{7}{3} = -2\frac{1}{3} \approx -2.333333333$$

Short Steps Using Substitution

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

$$y=-2x-z$$

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

$$x-\left(-2x-z\right)-z=4 x+2\left(-2x-z\right)-4z=10$$

Solve these equations for x and z respectively.

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

Substitute \frac{4}{3} for x in the equation z=-\frac{1}{2}x-\frac{5}{3}.

$$z=-\frac{1}{2}\times \frac{4}{3}-\frac{5}{3}$$

Calculate z from z=-\frac{1}{2}\times \frac{4}{3}-\frac{5}{3}.

$$z=-\frac{7}{3}$$

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

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

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

$$y=-\frac{1}{3}$$

The system is now solved.

$$x=\frac{4}{3} y=-\frac{1}{3} z=-\frac{7}{3}$$