Solve {l}{2x+y=2}{z=2y+6y-7}{text{Solvefor}atext{where}}{a=z}

$$\left. \begin{array} { l } { 2 x + y = 2 }\\ { z = 2 y + 6 y - 7 }\\ { \text{Solve for } a \text{ where} } \\ { a = z } \end{array} \right.$$

Solve for x, y, z

$$x=\frac{9-a}{16}$$

$$y=\frac{a+7}{8}$$

$$z=a$$

Short Steps Using Substitution

Reorder the equations.

$$z=2y+6y-7 2x+y=2 a=z$$

Calculate z from z=2y+6y-7.

$$z=8y-7$$

Substitute 8y-7 for z in the equation a=z.

$$a=8y-7$$

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

$$x=-\frac{1}{2}y+1 y=\frac{7}{8}+\frac{1}{8}a$$

Substitute \frac{7}{8}+\frac{1}{8}a for y in the equation x=-\frac{1}{2}y+1.

$$x=-\frac{1}{2}\left(\frac{7}{8}+\frac{1}{8}a\right)+1$$

Calculate x from x=-\frac{1}{2}\left(\frac{7}{8}+\frac{1}{8}a\right)+1.

$$x=\frac{9}{16}-\frac{1}{16}a$$

Substitute \frac{7}{8}+\frac{1}{8}a for y in the equation z=8y-7.

$$z=8\left(\frac{7}{8}+\frac{1}{8}a\right)-7$$

Calculate z from z=8\left(\frac{7}{8}+\frac{1}{8}a\right)-7.

$$z=a$$

The system is now solved.

$$x=\frac{9}{16}-\frac{1}{16}a y=\frac{7}{8}+\frac{1}{8}a z=a$$