Solve {l}{x+6y=11}{z=3x-y-7}{text{Solvefor}atext{where}}{a=z}

$$\left. \begin{array} { l } { x + 6 y = 11 }\\ { z = 3 x - y - 7 }\\ { \text{Solve for } a \text{ where} } \\ { a = z } \end{array} \right.$$

Solve for x, y, z

$$x=\frac{6a+53}{19}$$

$$y=\frac{26-a}{19}$$

$$z=a$$

Short Steps Using Substitution

Reorder the equations.

$$z=3x-y-7 x+6y=11 a=z$$

Substitute 3x-y-7 for z in the equation a=z.

$$a=3x-y-7$$

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

$$y=-\frac{1}{6}x+\frac{11}{6} x=\frac{7}{3}+\frac{1}{3}y+\frac{1}{3}a$$

Substitute -\frac{1}{6}x+\frac{11}{6} for y in the equation x=\frac{7}{3}+\frac{1}{3}y+\frac{1}{3}a.

$$x=\frac{7}{3}+\frac{1}{3}\left(-\frac{1}{6}x+\frac{11}{6}\right)+\frac{1}{3}a$$

Solve x=\frac{7}{3}+\frac{1}{3}\left(-\frac{1}{6}x+\frac{11}{6}\right)+\frac{1}{3}a for x.

$$x=\frac{53}{19}+\frac{6}{19}a$$

Substitute \frac{53}{19}+\frac{6}{19}a for x in the equation y=-\frac{1}{6}x+\frac{11}{6}.

$$y=-\frac{1}{6}\left(\frac{53}{19}+\frac{6}{19}a\right)+\frac{11}{6}$$

Calculate y from y=-\frac{1}{6}\left(\frac{53}{19}+\frac{6}{19}a\right)+\frac{11}{6}.

$$y=\frac{26}{19}-\frac{1}{19}a$$

Substitute \frac{26}{19}-\frac{1}{19}a for y and \frac{53}{19}+\frac{6}{19}a for x in the equation z=3x-y-7.

$$z=3\left(\frac{53}{19}+\frac{6}{19}a\right)-\left(\frac{26}{19}-\frac{1}{19}a\right)-7$$

Calculate z from z=3\left(\frac{53}{19}+\frac{6}{19}a\right)-\left(\frac{26}{19}-\frac{1}{19}a\right)-7.

$$z=a$$

The system is now solved.

$$x=\frac{53}{19}+\frac{6}{19}a y=\frac{26}{19}-\frac{1}{19}a z=a$$