Solve {l}{x+y=1}{z=x-y}{text{Solvefor}atext{where}}{a=z}

$$\left. \begin{array} { l } { x + y = 1 }\\ { z = x - y }\\ { \text{Solve for } a \text{ where} } \\ { a = z } \end{array} \right.$$

Solve for x, y, z

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

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

$$z=a$$

Short Steps Using Substitution

Reorder the equations.

$$z=x-y x+y=1 a=z$$

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.

$$y=-x+1 x=a+y$$

Substitute -x+1 for y in the equation x=a+y.

$$x=a-x+1$$

Solve x=a-x+1 for x.

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

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

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

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

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

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

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

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

$$z=a$$

The system is now solved.

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