Solve {l}{x-y=2}{z=x+y-1}{text{Solvefor}atext{where}}{a=z}

$$\left. \begin{array} { l } { x - y = 2 }\\ { z = x + y - 1 }\\ { \text{Solve for } a \text{ where} } \\ { a = z } \end{array} \right.$$

Solve for x, y, z

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

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

$$z=a$$

Short Steps Using Substitution

Reorder the equations.

$$z=x+y-1 x-y=2 a=z$$

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

$$a=x+y-1$$

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

$$y=x-2 x=-y+a+1$$

Substitute x-2 for y in the equation x=-y+a+1.

$$x=-\left(x-2\right)+a+1$$

Solve x=-\left(x-2\right)+a+1 for x.

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

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

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

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

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

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

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

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

$$z=a$$

The system is now solved.

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