Solve {l}{2x+3y=1}{z=3x+2y-x}{text{Solvefor}atext{where}}{a=z}

$$\left. \begin{array} { l } { 2 x + 3 y = 1 }\\ { z = 3 x + 2 y - x }\\ { \text{Solve for } a \text{ where} } \\ { a = z } \end{array} \right.$$

Solve for x, y, z

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

$$y=1-a$$

$$z=a$$

Short Steps Using Substitution

Reorder the equations.

$$z=3x+2y-x 2x+3y=1 a=z$$

Calculate z from z=3x+2y-x.

$$z=2x+2y$$

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

$$a=2x+2y$$

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

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

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

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

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

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

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

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

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

$$y=1-a$$

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

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

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

$$z=a$$

The system is now solved.

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