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

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

Solve for x, y, z

$$x=\frac{6-a}{7}$$

$$y=\frac{5a-9}{7}$$

$$z=a$$

Short Steps Using Substitution

Reorder the equations.

$$z=3x+2y 5x+y=3 a=z$$

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

$$a=3x+2y$$

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

$$y=-5x+3 x=\frac{1}{3}a-\frac{2}{3}y$$

Substitute -5x+3 for y in the equation x=\frac{1}{3}a-\frac{2}{3}y.

$$x=\frac{1}{3}a-\frac{2}{3}\left(-5x+3\right)$$

Solve x=\frac{1}{3}a-\frac{2}{3}\left(-5x+3\right) for x.

$$x=\frac{6}{7}-\frac{1}{7}a$$

Substitute \frac{6}{7}-\frac{1}{7}a for x in the equation y=-5x+3.

$$y=-5\left(\frac{6}{7}-\frac{1}{7}a\right)+3$$

Calculate y from y=-5\left(\frac{6}{7}-\frac{1}{7}a\right)+3.

$$y=-\frac{9}{7}+\frac{5}{7}a$$

Substitute -\frac{9}{7}+\frac{5}{7}a for y and \frac{6}{7}-\frac{1}{7}a for x in the equation z=3x+2y.

$$z=3\left(\frac{6}{7}-\frac{1}{7}a\right)+2\left(-\frac{9}{7}+\frac{5}{7}a\right)$$

Calculate z from z=3\left(\frac{6}{7}-\frac{1}{7}a\right)+2\left(-\frac{9}{7}+\frac{5}{7}a\right).

$$z=a$$

The system is now solved.

$$x=\frac{6}{7}-\frac{1}{7}a y=-\frac{9}{7}+\frac{5}{7}a z=a$$