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

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

Solve for x, y, z

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

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

$$z=a$$

Short Steps Using Substitution

Reorder the equations.

$$z=x-\left(x-y\right)+x x-y=y a=z$$

Calculate z from z=x-\left(x-y\right)+x.

$$z=x+y$$

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

$$a=x+y$$

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

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

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

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

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

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

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

$$y=\frac{1}{2}\times \frac{2}{3}a$$

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

$$y=\frac{1}{3}a$$

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

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

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

$$z=a$$

The system is now solved.

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