Solve {l}{10=3*5x+3y}{z=2x+y}{text{Solvefor}atext{where}}{a=z}

$$\left. \begin{array} { l } { 10 = 3 \cdot 5 x + 3 y }\\ { z = 2 x + y }\\ { \text{Solve for } a \text{ where} } \\ { a = z } \end{array} \right.$$

Solve for x, y, z

$$x=-\frac{a}{3}+\frac{10}{9}$$

$$y=\frac{5a}{3}-\frac{20}{9}$$

$$z=a$$

Short Steps Using Substitution

Reorder the equations.

$$z=2x+y 10=3\times 5x+3y a=z$$

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

$$a=2x+y$$

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

$$y=\frac{10}{3}-5x x=\frac{1}{2}a-\frac{1}{2}y$$

Substitute \frac{10}{3}-5x for y in the equation x=\frac{1}{2}a-\frac{1}{2}y.

$$x=\frac{1}{2}a-\frac{1}{2}\left(\frac{10}{3}-5x\right)$$

Solve x=\frac{1}{2}a-\frac{1}{2}\left(\frac{10}{3}-5x\right) for x.

$$x=\frac{10}{9}-\frac{1}{3}a$$

Substitute \frac{10}{9}-\frac{1}{3}a for x in the equation y=\frac{10}{3}-5x.

$$y=\frac{10}{3}-5\left(\frac{10}{9}-\frac{1}{3}a\right)$$

Calculate y from y=\frac{10}{3}-5\left(\frac{10}{9}-\frac{1}{3}a\right).

$$y=-\frac{20}{9}+\frac{5}{3}a$$

Substitute -\frac{20}{9}+\frac{5}{3}a for y and \frac{10}{9}-\frac{1}{3}a for x in the equation z=2x+y.

$$z=2\left(\frac{10}{9}-\frac{1}{3}a\right)-\frac{20}{9}+\frac{5}{3}a$$

Calculate z from z=2\left(\frac{10}{9}-\frac{1}{3}a\right)-\frac{20}{9}+\frac{5}{3}a.

$$z=a$$

The system is now solved.

$$x=\frac{10}{9}-\frac{1}{3}a y=-\frac{20}{9}+\frac{5}{3}a z=a$$