Solve {l}{2x+2y-z=7}{-2x-y+2z=-6}{-4x-2y+3z=-9}

$$\left. \begin{array}{l}{ 2 x + 2 y - z = 7 }\\{ - 2 x - y + 2 z = - 6 }\\{ - 4 x - 2 y + 3 z = - 9 }\end{array} \right.$$

Solve for x, y, z

$$x=-2$$

$$y=4$$

$$z=-3$$

Short Steps Using Substitution

Solve 2x+2y-z=7 for z.

$$z=2x+2y-7$$

Substitute 2x+2y-7 for z in the second and third equation.

$$-2x-y+2\left(2x+2y-7\right)=-6 -4x-2y+3\left(2x+2y-7\right)=-9$$

Solve these equations for y and x respectively.

$$y=\frac{8}{3}-\frac{2}{3}x x=6-2y$$

Substitute \frac{8}{3}-\frac{2}{3}x for y in the equation x=6-2y.

$$x=6-2\left(\frac{8}{3}-\frac{2}{3}x\right)$$

Solve x=6-2\left(\frac{8}{3}-\frac{2}{3}x\right) for x.

$$x=-2$$

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

$$y=\frac{8}{3}-\frac{2}{3}\left(-2\right)$$

Calculate y from y=\frac{8}{3}-\frac{2}{3}\left(-2\right).

$$y=4$$

Substitute 4 for y and -2 for x in the equation z=2x+2y-7.

$$z=2\left(-2\right)+2\times 4-7$$

Calculate z from z=2\left(-2\right)+2\times 4-7.

$$z=-3$$

The system is now solved.

$$x=-2 y=4 z=-3$$