Solve {l}{2x+2y=2}{z=4x+y-1}{text{Solvefor}atext{where}}{a=z}

$$\left. \begin{array} { l } { 2 x + 2 y = 2 }\\ { z = 4 x + y - 1 }\\ { \text{Solve for } a \text{ where} } \\ { a = z } \end{array} \right.$$

Solve for x, y, z

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

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

$$z=a$$

Short Steps Using Substitution

Reorder the equations.

$$z=4x+y-1 2x+2y=2 a=z$$

Substitute 4x+y-1 for z in the equation a=z.

$$a=4x+y-1$$

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

$$y=1-x x=\frac{1}{4}-\frac{1}{4}y+\frac{1}{4}a$$

Substitute 1-x for y in the equation x=\frac{1}{4}-\frac{1}{4}y+\frac{1}{4}a.

$$x=\frac{1}{4}-\frac{1}{4}\left(1-x\right)+\frac{1}{4}a$$

Solve x=\frac{1}{4}-\frac{1}{4}\left(1-x\right)+\frac{1}{4}a for x.

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

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

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

Substitute 1-\frac{1}{3}a for y and \frac{1}{3}a for x in the equation z=4x+y-1.

$$z=4\times \frac{1}{3}a+1-\frac{1}{3}a-1$$

Calculate z from z=4\times \frac{1}{3}a+1-\frac{1}{3}a-1.

$$z=a$$

The system is now solved.

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