Solve {l}{54=6x+6y}{z=50-6x+7y}{text{Solvefor}atext{where}}{a=z}

$$\left. \begin{array} { l } { 54 = 6 x + 6 y }\\ { z = 50 - 6 x + 7 y }\\ { \text{Solve for } a \text{ where} } \\ { a = z } \end{array} \right.$$

Solve for x, y, z

$$x=\frac{113-a}{13}$$

$$y=\frac{a+4}{13}$$

$$z=a$$

Short Steps Using Substitution

Reorder the equations.

$$z=50-6x+7y 54=6x+6y a=z$$

Substitute 50-6x+7y for z in the equation a=z.

$$a=50-6x+7y$$

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

$$y=9-x x=\frac{25}{3}-\frac{1}{6}a+\frac{7}{6}y$$

Substitute 9-x for y in the equation x=\frac{25}{3}-\frac{1}{6}a+\frac{7}{6}y.

$$x=\frac{25}{3}-\frac{1}{6}a+\frac{7}{6}\left(9-x\right)$$

Solve x=\frac{25}{3}-\frac{1}{6}a+\frac{7}{6}\left(9-x\right) for x.

$$x=\frac{113}{13}-\frac{1}{13}a$$

Substitute \frac{113}{13}-\frac{1}{13}a for x in the equation y=9-x.

$$y=9-\left(\frac{113}{13}-\frac{1}{13}a\right)$$

Calculate y from y=9-\left(\frac{113}{13}-\frac{1}{13}a\right).

$$y=\frac{4}{13}+\frac{1}{13}a$$

Substitute \frac{4}{13}+\frac{1}{13}a for y and \frac{113}{13}-\frac{1}{13}a for x in the equation z=50-6x+7y.

$$z=50-6\left(\frac{113}{13}-\frac{1}{13}a\right)+7\left(\frac{4}{13}+\frac{1}{13}a\right)$$

Calculate z from z=50-6\left(\frac{113}{13}-\frac{1}{13}a\right)+7\left(\frac{4}{13}+\frac{1}{13}a\right).

$$z=a$$

The system is now solved.

$$x=\frac{113}{13}-\frac{1}{13}a y=\frac{4}{13}+\frac{1}{13}a z=a$$