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

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

Solve for x, y, z

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

$$y=\frac{a-6}{2}$$

$$z=a$$

Short Steps Using Substitution

Reorder the equations.

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

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.

$$y=x-6 x=a-y$$

Substitute x-6 for y in the equation x=a-y.

$$x=a-\left(x-6\right)$$

Solve x=a-\left(x-6\right) for x.

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

Substitute 3+\frac{1}{2}a for x in the equation y=x-6.

$$y=3+\frac{1}{2}a-6$$

Calculate y from y=3+\frac{1}{2}a-6.

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

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

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

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

$$z=a$$

The system is now solved.

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