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

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

Solve for x, y, z

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

$$y=\frac{a+8}{3}$$

$$z=a$$

Short Steps Using Substitution

Reorder the equations.

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

Substitute -2x-1 for z in the equation a=z.

$$a=-2x-1$$

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

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

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

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

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

$$y=\frac{8}{3}+\frac{1}{3}a$$

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

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

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

$$z=a$$

The system is now solved.

$$x=-\frac{1}{2}-\frac{1}{2}a y=\frac{8}{3}+\frac{1}{3}a z=a$$