Solve {l}{5x+6y=13}{z=7x+5y}{text{Solvefor}atext{where}}{a=z}

$$\left. \begin{array} { l } { 5 x + 6 y = 13 }\\ { z = 7 x + 5 y }\\ { \text{Solve for } a \text{ where} } \\ { a = z } \end{array} \right.$$

Solve for x, y, z

$$x=\frac{6a-65}{17}$$

$$y=\frac{91-5a}{17}$$

$$z=a$$

Short Steps Using Substitution

Reorder the equations.

$$z=7x+5y 5x+6y=13 a=z$$

Substitute 7x+5y for z in the equation a=z.

$$a=7x+5y$$

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

$$y=-\frac{5}{6}x+\frac{13}{6} x=\frac{1}{7}a-\frac{5}{7}y$$

Substitute -\frac{5}{6}x+\frac{13}{6} for y in the equation x=\frac{1}{7}a-\frac{5}{7}y.

$$x=\frac{1}{7}a-\frac{5}{7}\left(-\frac{5}{6}x+\frac{13}{6}\right)$$

Solve x=\frac{1}{7}a-\frac{5}{7}\left(-\frac{5}{6}x+\frac{13}{6}\right) for x.

$$x=-\frac{65}{17}+\frac{6}{17}a$$

Substitute -\frac{65}{17}+\frac{6}{17}a for x in the equation y=-\frac{5}{6}x+\frac{13}{6}.

$$y=-\frac{5}{6}\left(-\frac{65}{17}+\frac{6}{17}a\right)+\frac{13}{6}$$

Calculate y from y=-\frac{5}{6}\left(-\frac{65}{17}+\frac{6}{17}a\right)+\frac{13}{6}.

$$y=\frac{91}{17}-\frac{5}{17}a$$

Substitute \frac{91}{17}-\frac{5}{17}a for y and -\frac{65}{17}+\frac{6}{17}a for x in the equation z=7x+5y.

$$z=7\left(-\frac{65}{17}+\frac{6}{17}a\right)+5\left(\frac{91}{17}-\frac{5}{17}a\right)$$

Calculate z from z=7\left(-\frac{65}{17}+\frac{6}{17}a\right)+5\left(\frac{91}{17}-\frac{5}{17}a\right).

$$z=a$$

The system is now solved.

$$x=-\frac{65}{17}+\frac{6}{17}a y=\frac{91}{17}-\frac{5}{17}a z=a$$