Solve {l}{9x-4y=8}{z=11x+4y}{text{Solvefor}atext{where}}{a=z}

$$\left. \begin{array} { l } { 9 x - 4 y = 8 }\\ { z = 11 x + 4 y }\\ { \text{Solve for } a \text{ where} } \\ { a = z } \end{array} \right.$$

Solve for x, y, z

$$x=\frac{a}{20}+\frac{2}{5}$$

$$y=\frac{9a}{80}-\frac{11}{10}$$

$$z=a$$

Short Steps Using Substitution

Reorder the equations.

$$z=11x+4y 9x-4y=8 a=z$$

Substitute 11x+4y for z in the equation a=z.

$$a=11x+4y$$

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

$$y=\frac{9}{4}x-2 x=\frac{1}{11}a-\frac{4}{11}y$$

Substitute \frac{9}{4}x-2 for y in the equation x=\frac{1}{11}a-\frac{4}{11}y.

$$x=\frac{1}{11}a-\frac{4}{11}\left(\frac{9}{4}x-2\right)$$

Solve x=\frac{1}{11}a-\frac{4}{11}\left(\frac{9}{4}x-2\right) for x.

$$x=\frac{2}{5}+\frac{1}{20}a$$

Substitute \frac{2}{5}+\frac{1}{20}a for x in the equation y=\frac{9}{4}x-2.

$$y=\frac{9}{4}\left(\frac{2}{5}+\frac{1}{20}a\right)-2$$

Calculate y from y=\frac{9}{4}\left(\frac{2}{5}+\frac{1}{20}a\right)-2.

$$y=-\frac{11}{10}+\frac{9}{80}a$$

Substitute -\frac{11}{10}+\frac{9}{80}a for y and \frac{2}{5}+\frac{1}{20}a for x in the equation z=11x+4y.

$$z=11\left(\frac{2}{5}+\frac{1}{20}a\right)+4\left(-\frac{11}{10}+\frac{9}{80}a\right)$$

Calculate z from z=11\left(\frac{2}{5}+\frac{1}{20}a\right)+4\left(-\frac{11}{10}+\frac{9}{80}a\right).

$$z=a$$

The system is now solved.

$$x=\frac{2}{5}+\frac{1}{20}a y=-\frac{11}{10}+\frac{9}{80}a z=a$$