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

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

Solve for x, y, z

$$x=\frac{3a+4}{7}$$

$$y=\frac{2-2a}{7}$$

$$z=a$$

Short Steps Using Substitution

Reorder the equations.

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

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

$$a=x-2y$$

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

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

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

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

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

$$x=\frac{4}{7}+\frac{3}{7}a$$

Substitute \frac{4}{7}+\frac{3}{7}a for x in the equation y=\frac{2}{3}-\frac{2}{3}x.

$$y=\frac{2}{3}-\frac{2}{3}\left(\frac{4}{7}+\frac{3}{7}a\right)$$

Calculate y from y=\frac{2}{3}-\frac{2}{3}\left(\frac{4}{7}+\frac{3}{7}a\right).

$$y=\frac{2}{7}-\frac{2}{7}a$$

Substitute \frac{2}{7}-\frac{2}{7}a for y and \frac{4}{7}+\frac{3}{7}a for x in the equation z=x-2y.

$$z=\frac{4}{7}+\frac{3}{7}a-2\left(\frac{2}{7}-\frac{2}{7}a\right)$$

Calculate z from z=\frac{4}{7}+\frac{3}{7}a-2\left(\frac{2}{7}-\frac{2}{7}a\right).

$$z=a$$

The system is now solved.

$$x=\frac{4}{7}+\frac{3}{7}a y=\frac{2}{7}-\frac{2}{7}a z=a$$