Solve {l}{30y-2x=3y}{z=2x+3y}{text{Solvefor}atext{where}}{a=z}

$$\left. \begin{array} { l } { 30 y - 2 x = 3 y }\\ { z = 2 x + 3 y }\\ { \text{Solve for } a \text{ where} } \\ { a = z } \end{array} \right.$$

Solve for x, y, z

$$x=\frac{9a}{20}$$

$$y=\frac{a}{30}$$

$$z=a$$

Short Steps Using Substitution

Reorder the equations.

$$z=2x+3y 30y-2x=3y a=z$$

Substitute 2x+3y for z in the equation a=z.

$$a=2x+3y$$

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

$$y=\frac{2}{27}x x=\frac{1}{2}a-\frac{3}{2}y$$

Substitute \frac{2}{27}x for y in the equation x=\frac{1}{2}a-\frac{3}{2}y.

$$x=\frac{1}{2}a-\frac{3}{2}\times \frac{2}{27}x$$

Solve x=\frac{1}{2}a-\frac{3}{2}\times \frac{2}{27}x for x.

$$x=\frac{9}{20}a$$

Substitute \frac{9}{20}a for x in the equation y=\frac{2}{27}x.

$$y=\frac{2}{27}\times \frac{9}{20}a$$

Calculate y from y=\frac{2}{27}\times \frac{9}{20}a.

$$y=\frac{1}{30}a$$

Substitute \frac{1}{30}a for y and \frac{9}{20}a for x in the equation z=2x+3y.

$$z=2\times \frac{9}{20}a+3\times \frac{1}{30}a$$

Calculate z from z=2\times \frac{9}{20}a+3\times \frac{1}{30}a.

$$z=a$$

The system is now solved.

$$x=\frac{9}{20}a y=\frac{1}{30}a z=a$$