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

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

Solve for x, y, z

$$x=\frac{a+2}{3}$$

$$y=\frac{a-4}{3}$$

$$z=a$$

Short Steps Using Substitution

Reorder the equations.

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

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

$$a=2x+y$$

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

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

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

$$x=\frac{1}{2}a-\frac{1}{2}\left(x-2\right)$$

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

$$x=\frac{2}{3}+\frac{1}{3}a$$

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

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

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

$$y=-\frac{4}{3}+\frac{1}{3}a$$

Substitute -\frac{4}{3}+\frac{1}{3}a for y and \frac{2}{3}+\frac{1}{3}a for x in the equation z=2x+y.

$$z=2\left(\frac{2}{3}+\frac{1}{3}a\right)-\frac{4}{3}+\frac{1}{3}a$$

Calculate z from z=2\left(\frac{2}{3}+\frac{1}{3}a\right)-\frac{4}{3}+\frac{1}{3}a.

$$z=a$$

The system is now solved.

$$x=\frac{2}{3}+\frac{1}{3}a y=-\frac{4}{3}+\frac{1}{3}a z=a$$