Solve {l}{2y+3x=7}{z=-3x+1}{text{Solvefor}atext{where}}{a=z}

$$\left. \begin{array} { l } { 2 y + 3 x = 7 }\\ { z = -3 x + 1 }\\ { \text{Solve for } a \text{ where} } \\ { a = z } \end{array} \right.$$

Solve for x, y, z

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

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

$$z=a$$

Short Steps Using Substitution

Reorder the equations.

$$z=-3x+1 2y+3x=7 a=z$$

Substitute -3x+1 for z in the equation a=z.

$$a=-3x+1$$

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

$$y=-\frac{3}{2}x+\frac{7}{2} x=\frac{1}{3}-\frac{1}{3}a$$

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

$$y=-\frac{3}{2}\left(\frac{1}{3}-\frac{1}{3}a\right)+\frac{7}{2}$$

Calculate y from y=-\frac{3}{2}\left(\frac{1}{3}-\frac{1}{3}a\right)+\frac{7}{2}.

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

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

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

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

$$z=a$$

The system is now solved.

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