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

$$x=-2$$

$$y=3$$

$$z=6$$

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.

$$x+y+2y=7 3x+6y-2\times 2y=0$$

Solve these equations for y and x respectively.

$$y=-\frac{1}{3}x+\frac{7}{3} x=-\frac{2}{3}y$$

Substitute -\frac{1}{3}x+\frac{7}{3} for y in the equation x=-\frac{2}{3}y.

$$x=-\frac{2}{3}\left(-\frac{1}{3}x+\frac{7}{3}\right)$$

Solve x=-\frac{2}{3}\left(-\frac{1}{3}x+\frac{7}{3}\right) for x.

$$x=-2$$

Substitute -2 for x in the equation y=-\frac{1}{3}x+\frac{7}{3}.

$$y=-\frac{1}{3}\left(-2\right)+\frac{7}{3}$$

Calculate y from y=-\frac{1}{3}\left(-2\right)+\frac{7}{3}.

$$y=3$$

Substitute 3 for y in the equation z=2y.

$$z=2\times 3$$

Calculate z from z=2\times 3.

$$z=6$$

The system is now solved.

$$x=-2 y=3 z=6$$