Solve {l}{2x+2y+z=4}{2x+y+2z=7}{x+2y+2z=-6}

$$\left\{ \begin{array} { l } { 2 x + 2 y + z = 4 } \\ { 2 x + y + 2 z = 7 } \\ { x + 2 y + 2 z = - 6 } \end{array} \right.$$

Solve for x, y, z

$$x=8$$

$$y=-5$$

$$z=-2$$

Short Steps Using Substitution

Solve 2x+2y+z=4 for z.

$$z=4-2x-2y$$

Substitute 4-2x-2y for z in the second and third equation.

$$2x+y+2\left(4-2x-2y\right)=7 x+2y+2\left(4-2x-2y\right)=-6$$

Solve these equations for y and x respectively.

$$y=-\frac{2}{3}x+\frac{1}{3} x=\frac{14}{3}-\frac{2}{3}y$$

Substitute -\frac{2}{3}x+\frac{1}{3} for y in the equation x=\frac{14}{3}-\frac{2}{3}y.

$$x=\frac{14}{3}-\frac{2}{3}\left(-\frac{2}{3}x+\frac{1}{3}\right)$$

Solve x=\frac{14}{3}-\frac{2}{3}\left(-\frac{2}{3}x+\frac{1}{3}\right) for x.

$$x=8$$

Substitute 8 for x in the equation y=-\frac{2}{3}x+\frac{1}{3}.

$$y=-\frac{2}{3}\times 8+\frac{1}{3}$$

Calculate y from y=-\frac{2}{3}\times 8+\frac{1}{3}.

$$y=-5$$

Substitute -5 for y and 8 for x in the equation z=4-2x-2y.

$$z=4-2\times 8-2\left(-5\right)$$

Calculate z from z=4-2\times 8-2\left(-5\right).

$$z=-2$$

The system is now solved.

$$x=8 y=-5 z=-2$$