Solve {l}{2y=x+6}{z=4y-5x+6}{text{Solvefor}atext{where}}{a=z}

$$\left. \begin{array} { l } { 2 y = x + 6 }\\ { z = 4 y - 5 x + 6 }\\ { \text{Solve for } a \text{ where} } \\ { a = z } \end{array} \right.$$

Solve for x, y, z

$$x=-\frac{a}{3}+6$$

$$y=-\frac{a}{6}+6$$

$$z=a$$

Short Steps Using Substitution

Reorder the equations.

$$z=4y-5x+6 2y=x+6 a=z$$

Substitute 4y-5x+6 for z in the equation a=z.

$$a=4y-5x+6$$

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

$$y=\frac{1}{2}x+3 x=\frac{4}{5}y-\frac{1}{5}a+\frac{6}{5}$$

Substitute \frac{1}{2}x+3 for y in the equation x=\frac{4}{5}y-\frac{1}{5}a+\frac{6}{5}.

$$x=\frac{4}{5}\left(\frac{1}{2}x+3\right)-\frac{1}{5}a+\frac{6}{5}$$

Solve x=\frac{4}{5}\left(\frac{1}{2}x+3\right)-\frac{1}{5}a+\frac{6}{5} for x.

$$x=6-\frac{1}{3}a$$

Substitute 6-\frac{1}{3}a for x in the equation y=\frac{1}{2}x+3.

$$y=\frac{1}{2}\left(6-\frac{1}{3}a\right)+3$$

Calculate y from y=\frac{1}{2}\left(6-\frac{1}{3}a\right)+3.

$$y=6-\frac{1}{6}a$$

Substitute 6-\frac{1}{6}a for y and 6-\frac{1}{3}a for x in the equation z=4y-5x+6.

$$z=4\left(6-\frac{1}{6}a\right)-5\left(6-\frac{1}{3}a\right)+6$$

Calculate z from z=4\left(6-\frac{1}{6}a\right)-5\left(6-\frac{1}{3}a\right)+6.

$$z=a$$

The system is now solved.

$$x=6-\frac{1}{3}a y=6-\frac{1}{6}a z=a$$