Solve {l}{x+3y=6}{z=4x-6y}{text{Solvefor}atext{where}}{a=z}

$$\left. \begin{array} { l } { x + 3 y = 6 }\\ { z = 4 x - 6 y }\\ { \text{Solve for } a \text{ where} } \\ { a = z } \end{array} \right.$$

Solve for x, y, z

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

$$y=-\frac{a}{18}+\frac{4}{3}$$

$$z=a$$

Short Steps Using Substitution

Reorder the equations.

$$z=4x-6y x+3y=6 a=z$$

Substitute 4x-6y for z in the equation a=z.

$$a=4x-6y$$

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

$$y=-\frac{1}{3}x+2 x=\frac{1}{4}a+\frac{3}{2}y$$

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

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

Solve x=\frac{1}{4}a+\frac{3}{2}\left(-\frac{1}{3}x+2\right) for x.

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

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

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

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

$$y=\frac{4}{3}-\frac{1}{18}a$$

Substitute \frac{4}{3}-\frac{1}{18}a for y and 2+\frac{1}{6}a for x in the equation z=4x-6y.

$$z=4\left(2+\frac{1}{6}a\right)-6\left(\frac{4}{3}-\frac{1}{18}a\right)$$

Calculate z from z=4\left(2+\frac{1}{6}a\right)-6\left(\frac{4}{3}-\frac{1}{18}a\right).

$$z=a$$

The system is now solved.

$$x=2+\frac{1}{6}a y=\frac{4}{3}-\frac{1}{18}a z=a$$