Solve {l}{6x-7y=-12}{z=6x-6y+6}{text{Solvefor}atext{where}}{a=z}

$$\left. \begin{array} { l } { 6 x - 7 y = -12 }\\ { z = 6 x - 6 y + 6 }\\ { \text{Solve for } a \text{ where} } \\ { a = z } \end{array} \right.$$

Solve for x, y, z

$$x=\frac{7a}{6}+5$$

$$y=a+6$$

$$z=a$$

Short Steps Using Substitution

Reorder the equations.

$$z=6x-6y+6 6x-7y=-12 a=z$$

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

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

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

$$y=\frac{12}{7}+\frac{6}{7}x x=-1+\frac{1}{6}a+y$$

Substitute \frac{12}{7}+\frac{6}{7}x for y in the equation x=-1+\frac{1}{6}a+y.

$$x=-1+\frac{1}{6}a+\frac{12}{7}+\frac{6}{7}x$$

Solve x=-1+\frac{1}{6}a+\frac{12}{7}+\frac{6}{7}x for x.

$$x=5+\frac{7}{6}a$$

Substitute 5+\frac{7}{6}a for x in the equation y=\frac{12}{7}+\frac{6}{7}x.

$$y=\frac{12}{7}+\frac{6}{7}\left(5+\frac{7}{6}a\right)$$

Calculate y from y=\frac{12}{7}+\frac{6}{7}\left(5+\frac{7}{6}a\right).

$$y=6+a$$

Substitute 6+a for y and 5+\frac{7}{6}a for x in the equation z=6x-6y+6.

$$z=6\left(5+\frac{7}{6}a\right)-6\left(6+a\right)+6$$

Calculate z from z=6\left(5+\frac{7}{6}a\right)-6\left(6+a\right)+6.

$$z=a$$

The system is now solved.

$$x=5+\frac{7}{6}a y=6+a z=a$$