Solve {l}{5x+2y=9}{z=x+y}{text{Solvefor}atext{where}}{a=z}

$$\left. \begin{array} { l } { 5 x + 2 y = 9 }\\ { z = x + y }\\ { \text{Solve for } a \text{ where} } \\ { a = z } \end{array} \right.$$

Solve for x, y, z

$$x=-\frac{2a}{3}+3$$

$$y=\frac{5a}{3}-3$$

$$z=a$$

Short Steps Using Substitution

Reorder the equations.

$$z=x+y 5x+2y=9 a=z$$

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

$$a=x+y$$

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

$$y=-\frac{5}{2}x+\frac{9}{2} x=a-y$$

Substitute -\frac{5}{2}x+\frac{9}{2} for y in the equation x=a-y.

$$x=a-\left(-\frac{5}{2}x+\frac{9}{2}\right)$$

Solve x=a-\left(-\frac{5}{2}x+\frac{9}{2}\right) for x.

$$x=3-\frac{2}{3}a$$

Substitute 3-\frac{2}{3}a for x in the equation y=-\frac{5}{2}x+\frac{9}{2}.

$$y=-\frac{5}{2}\left(3-\frac{2}{3}a\right)+\frac{9}{2}$$

Calculate y from y=-\frac{5}{2}\left(3-\frac{2}{3}a\right)+\frac{9}{2}.

$$y=-3+\frac{5}{3}a$$

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

$$z=3-\frac{2}{3}a-3+\frac{5}{3}a$$

Calculate z from z=3-\frac{2}{3}a-3+\frac{5}{3}a.

$$z=a$$

The system is now solved.

$$x=3-\frac{2}{3}a y=-3+\frac{5}{3}a z=a$$