Solve {l}{2x+3y=10}{z=21+2y}{text{Solvefor}atext{where}}{a=z}

$$\left. \begin{array} { l } { 2 x + 3 y = 10 }\\ { z = 21 + 2 y }\\ { \text{Solve for } a \text{ where} } \\ { a = z } \end{array} \right.$$

Solve for x, y, z

$$x=\frac{83-3a}{4}$$

$$y=\frac{a-21}{2}$$

$$z=a$$

Short Steps Using Substitution

Reorder the equations.

$$z=21+2y 2x+3y=10 a=z$$

Substitute 21+2y for z in the equation a=z.

$$a=21+2y$$

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

$$x=-\frac{3}{2}y+5 y=-\frac{21}{2}+\frac{1}{2}a$$

Substitute -\frac{21}{2}+\frac{1}{2}a for y in the equation x=-\frac{3}{2}y+5.

$$x=-\frac{3}{2}\left(-\frac{21}{2}+\frac{1}{2}a\right)+5$$

Calculate x from x=-\frac{3}{2}\left(-\frac{21}{2}+\frac{1}{2}a\right)+5.

$$x=\frac{83}{4}-\frac{3}{4}a$$

Substitute -\frac{21}{2}+\frac{1}{2}a for y in the equation z=21+2y.

$$z=21+2\left(-\frac{21}{2}+\frac{1}{2}a\right)$$

Calculate z from z=21+2\left(-\frac{21}{2}+\frac{1}{2}a\right).

$$z=a$$

The system is now solved.

$$x=\frac{83}{4}-\frac{3}{4}a y=-\frac{21}{2}+\frac{1}{2}a z=a$$