Solve {l}{4x=-7y+31}{z=9x-22y+19}{text{Solvefor}atext{where}}{a=z}

$$\left. \begin{array} { l } { 4 x = -7 y + 31 }\\ { z = 9 x - 22 y + 19 }\\ { \text{Solve for } a \text{ where} } \\ { a = z } \end{array} \right.$$

Solve for x, y, z

$$x=\frac{7a+549}{151}$$

$$y=\frac{355-4a}{151}$$

$$z=a$$

Short Steps Using Substitution

Reorder the equations.

$$z=9x-22y+19 4x=-7y+31 a=z$$

Substitute 9x-22y+19 for z in the equation a=z.

$$a=9x-22y+19$$

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

$$y=-\frac{4}{7}x+\frac{31}{7} x=-\frac{19}{9}+\frac{22}{9}y+\frac{1}{9}a$$

Substitute -\frac{4}{7}x+\frac{31}{7} for y in the equation x=-\frac{19}{9}+\frac{22}{9}y+\frac{1}{9}a.

$$x=-\frac{19}{9}+\frac{22}{9}\left(-\frac{4}{7}x+\frac{31}{7}\right)+\frac{1}{9}a$$

Solve x=-\frac{19}{9}+\frac{22}{9}\left(-\frac{4}{7}x+\frac{31}{7}\right)+\frac{1}{9}a for x.

$$x=\frac{549}{151}+\frac{7}{151}a$$

Substitute \frac{549}{151}+\frac{7}{151}a for x in the equation y=-\frac{4}{7}x+\frac{31}{7}.

$$y=-\frac{4}{7}\left(\frac{549}{151}+\frac{7}{151}a\right)+\frac{31}{7}$$

Calculate y from y=-\frac{4}{7}\left(\frac{549}{151}+\frac{7}{151}a\right)+\frac{31}{7}.

$$y=\frac{355}{151}-\frac{4}{151}a$$

Substitute \frac{355}{151}-\frac{4}{151}a for y and \frac{549}{151}+\frac{7}{151}a for x in the equation z=9x-22y+19.

$$z=9\left(\frac{549}{151}+\frac{7}{151}a\right)-22\left(\frac{355}{151}-\frac{4}{151}a\right)+19$$

Calculate z from z=9\left(\frac{549}{151}+\frac{7}{151}a\right)-22\left(\frac{355}{151}-\frac{4}{151}a\right)+19.

$$z=a$$

The system is now solved.

$$x=\frac{549}{151}+\frac{7}{151}a y=\frac{355}{151}-\frac{4}{151}a z=a$$