Solve {l}{x+y=0}{z=x-y}{text{Solvefor}atext{where}}{a=z}

$$\left. \begin{array} { l } { x + y = 0 }\\ { z = x - y }\\ { \text{Solve for } a \text{ where} } \\ { a = z } \end{array} \right.$$

Solve for x, y, z

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

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

$$z=a$$

Short Steps Using Substitution

Reorder the equations.

$$z=x-y x+y=0 a=z$$

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.

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

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

$$x=a-x$$

Solve x=a-x for x.

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

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

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

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

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

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

$$z=a$$

The system is now solved.

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