Solve {l}{5x+y=4}{z=x-y}{text{Solvefor}atext{where}}{a=z}

$$\left. \begin{array} { l } { 5 x + y = 4 }\\ { z = x - y }\\ { \text{Solve for } a \text{ where} } \\ { a = z } \end{array} \right.$$

Solve for x, y, z

$$x=\frac{a}{6}+\frac{2}{3}$$

$$y=-\frac{5a}{6}+\frac{2}{3}$$

$$z=a$$

Short Steps Using Substitution

Reorder the equations.

$$z=x-y 5x+y=4 a=z$$

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.

$$y=-5x+4 x=a+y$$

Substitute -5x+4 for y in the equation x=a+y.

$$x=a-5x+4$$

Solve x=a-5x+4 for x.

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

Substitute \frac{2}{3}+\frac{1}{6}a for x in the equation y=-5x+4.

$$y=-5\left(\frac{2}{3}+\frac{1}{6}a\right)+4$$

Calculate y from y=-5\left(\frac{2}{3}+\frac{1}{6}a\right)+4.

$$y=\frac{2}{3}-\frac{5}{6}a$$

Substitute \frac{2}{3}-\frac{5}{6}a for y and \frac{2}{3}+\frac{1}{6}a for x in the equation z=x-y.

$$z=\frac{2}{3}+\frac{1}{6}a-\left(\frac{2}{3}-\frac{5}{6}a\right)$$

Calculate z from z=\frac{2}{3}+\frac{1}{6}a-\left(\frac{2}{3}-\frac{5}{6}a\right).

$$z=a$$

The system is now solved.

$$x=\frac{2}{3}+\frac{1}{6}a y=\frac{2}{3}-\frac{5}{6}a z=a$$