Solve {l}{4x-5y=2}{z=12x+3y}{text{Solvefor}atext{where}}{a=z}

$$\left. \begin{array} { l } { 4 x - 5 y = 2 }\\ { z = 12 x + 3 y }\\ { \text{Solve for } a \text{ where} } \\ { a = z } \end{array} \right.$$

Solve for x, y, z

$$x=\frac{5a}{72}+\frac{1}{12}$$

$$y=\frac{a}{18}-\frac{1}{3}$$

$$z=a$$

Short Steps Using Substitution

Reorder the equations.

$$z=12x+3y 4x-5y=2 a=z$$

Substitute 12x+3y for z in the equation a=z.

$$a=12x+3y$$

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

$$y=-\frac{2}{5}+\frac{4}{5}x x=\frac{1}{12}a-\frac{1}{4}y$$

Substitute -\frac{2}{5}+\frac{4}{5}x for y in the equation x=\frac{1}{12}a-\frac{1}{4}y.

$$x=\frac{1}{12}a-\frac{1}{4}\left(-\frac{2}{5}+\frac{4}{5}x\right)$$

Solve x=\frac{1}{12}a-\frac{1}{4}\left(-\frac{2}{5}+\frac{4}{5}x\right) for x.

$$x=\frac{1}{12}+\frac{5}{72}a$$

Substitute \frac{1}{12}+\frac{5}{72}a for x in the equation y=-\frac{2}{5}+\frac{4}{5}x.

$$y=-\frac{2}{5}+\frac{4}{5}\left(\frac{1}{12}+\frac{5}{72}a\right)$$

Calculate y from y=-\frac{2}{5}+\frac{4}{5}\left(\frac{1}{12}+\frac{5}{72}a\right).

$$y=-\frac{1}{3}+\frac{1}{18}a$$

Substitute -\frac{1}{3}+\frac{1}{18}a for y and \frac{1}{12}+\frac{5}{72}a for x in the equation z=12x+3y.

$$z=12\left(\frac{1}{12}+\frac{5}{72}a\right)+3\left(-\frac{1}{3}+\frac{1}{18}a\right)$$

Calculate z from z=12\left(\frac{1}{12}+\frac{5}{72}a\right)+3\left(-\frac{1}{3}+\frac{1}{18}a\right).

$$z=a$$

The system is now solved.

$$x=\frac{1}{12}+\frac{5}{72}a y=-\frac{1}{3}+\frac{1}{18}a z=a$$