Solve {l}{x+y+z=7}{z=-2y}{3x+6y-2z=0}

$$\left\{ \begin{array} { l } { x + y + z = 7 } \\ { z = - 2 y } \\ { 3 x + 6 y - 2 z = 0 } \end{array} \right.$$

Solve for x, y, z

$$x = \frac{70}{13} = 5\frac{5}{13} \approx 5.384615385$$

$$y = -\frac{21}{13} = -1\frac{8}{13} \approx -1.615384615$$

$$z = \frac{42}{13} = 3\frac{3}{13} \approx 3.230769231$$

Short Steps Using Substitution

Reorder the equations.

$$z=-2y x+y+z=7 3x+6y-2z=0$$

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

$$x+y-2y=7 3x+6y-2\left(-2\right)y=0$$

Solve these equations for y and x respectively.

$$y=x-7 x=-\frac{10}{3}y$$

Substitute x-7 for y in the equation x=-\frac{10}{3}y.

$$x=-\frac{10}{3}\left(x-7\right)$$

Solve x=-\frac{10}{3}\left(x-7\right) for x.

$$x=\frac{70}{13}$$

Substitute \frac{70}{13} for x in the equation y=x-7.

$$y=\frac{70}{13}-7$$

Calculate y from y=\frac{70}{13}-7.

$$y=-\frac{21}{13}$$

Substitute -\frac{21}{13} for y in the equation z=-2y.

$$z=-2\left(-\frac{21}{13}\right)$$

Calculate z from z=-2\left(-\frac{21}{13}\right).

$$z=\frac{42}{13}$$

The system is now solved.

$$x=\frac{70}{13} y=-\frac{21}{13} z=\frac{42}{13}$$