Solve {l}{5y+x=2}{z=2x-y-7}{text{Solvefor}atext{where}}{a=z}

$$\left. \begin{array} { l } { 5 y + x = 2 }\\ { z = 2 x - y - 7 }\\ { \text{Solve for } a \text{ where} } \\ { a = z } \end{array} \right.$$

Solve for x, y, z

$$x=\frac{5a+37}{11}$$

$$y=\frac{-a-3}{11}$$

$$z=a$$

Short Steps Using Substitution

Reorder the equations.

$$z=2x-y-7 5y+x=2 a=z$$

Substitute 2x-y-7 for z in the equation a=z.

$$a=2x-y-7$$

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

$$y=-\frac{1}{5}x+\frac{2}{5} x=\frac{7}{2}+\frac{1}{2}y+\frac{1}{2}a$$

Substitute -\frac{1}{5}x+\frac{2}{5} for y in the equation x=\frac{7}{2}+\frac{1}{2}y+\frac{1}{2}a.

$$x=\frac{7}{2}+\frac{1}{2}\left(-\frac{1}{5}x+\frac{2}{5}\right)+\frac{1}{2}a$$

Solve x=\frac{7}{2}+\frac{1}{2}\left(-\frac{1}{5}x+\frac{2}{5}\right)+\frac{1}{2}a for x.

$$x=\frac{37}{11}+\frac{5}{11}a$$

Substitute \frac{37}{11}+\frac{5}{11}a for x in the equation y=-\frac{1}{5}x+\frac{2}{5}.

$$y=-\frac{1}{5}\left(\frac{37}{11}+\frac{5}{11}a\right)+\frac{2}{5}$$

Calculate y from y=-\frac{1}{5}\left(\frac{37}{11}+\frac{5}{11}a\right)+\frac{2}{5}.

$$y=-\frac{3}{11}-\frac{1}{11}a$$

Substitute -\frac{3}{11}-\frac{1}{11}a for y and \frac{37}{11}+\frac{5}{11}a for x in the equation z=2x-y-7.

$$z=2\left(\frac{37}{11}+\frac{5}{11}a\right)-\left(-\frac{3}{11}-\frac{1}{11}a\right)-7$$

Calculate z from z=2\left(\frac{37}{11}+\frac{5}{11}a\right)-\left(-\frac{3}{11}-\frac{1}{11}a\right)-7.

$$z=a$$

The system is now solved.

$$x=\frac{37}{11}+\frac{5}{11}a y=-\frac{3}{11}-\frac{1}{11}a z=a$$