Solve {l}{2x-y=-6}{z=-2x+2y}{text{Solvefor}atext{where}}{a=z}

$$\left. \begin{array} { l } { 2 x - y = -6 }\\ { z = -2 x + 2 y }\\ { \text{Solve for } a \text{ where} } \\ { a = z } \end{array} \right.$$

Solve for x, y, z

$$x=\frac{a-12}{2}$$

$$y=a-6$$

$$z=a$$

Short Steps Using Substitution

Reorder the equations.

$$z=-2x+2y 2x-y=-6 a=z$$

Substitute -2x+2y for z in the equation a=z.

$$a=-2x+2y$$

Solve the second equation for y and the third equation for x.

$$y=6+2x x=-\frac{1}{2}a+y$$

Substitute 6+2x for y in the equation x=-\frac{1}{2}a+y.

$$x=-\frac{1}{2}a+6+2x$$

Solve x=-\frac{1}{2}a+6+2x for x.

$$x=-6+\frac{1}{2}a$$

Substitute -6+\frac{1}{2}a for x in the equation y=6+2x.

$$y=6+2\left(-6+\frac{1}{2}a\right)$$

Calculate y from y=6+2\left(-6+\frac{1}{2}a\right).

$$y=-6+a$$

Substitute -6+a for y and -6+\frac{1}{2}a for x in the equation z=-2x+2y.

$$z=-2\left(-6+\frac{1}{2}a\right)+2\left(-6+a\right)$$

Calculate z from z=-2\left(-6+\frac{1}{2}a\right)+2\left(-6+a\right).

$$z=a$$

The system is now solved.

$$x=-6+\frac{1}{2}a y=-6+a z=a$$