Solve {l}{3x+5y=74}{z=2x+3y-46}{text{Solvefor}atext{where}}{a=z}

$$\left. \begin{array} { l } { 3 x + 5 y = 74 }\\ { z = 2 x + 3 y - 46 }\\ { \text{Solve for } a \text{ where} } \\ { a = z } \end{array} \right.$$

Solve for x, y, z

$$x=5a+8$$

$$y=10-3a$$

$$z=a$$

Short Steps Using Substitution

Reorder the equations.

$$z=2x+3y-46 3x+5y=74 a=z$$

Substitute 2x+3y-46 for z in the equation a=z.

$$a=2x+3y-46$$

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

$$y=-\frac{3}{5}x+\frac{74}{5} x=-\frac{3}{2}y+\frac{1}{2}a+23$$

Substitute -\frac{3}{5}x+\frac{74}{5} for y in the equation x=-\frac{3}{2}y+\frac{1}{2}a+23.

$$x=-\frac{3}{2}\left(-\frac{3}{5}x+\frac{74}{5}\right)+\frac{1}{2}a+23$$

Solve x=-\frac{3}{2}\left(-\frac{3}{5}x+\frac{74}{5}\right)+\frac{1}{2}a+23 for x.

$$x=8+5a$$

Substitute 8+5a for x in the equation y=-\frac{3}{5}x+\frac{74}{5}.

$$y=-\frac{3}{5}\left(8+5a\right)+\frac{74}{5}$$

Calculate y from y=-\frac{3}{5}\left(8+5a\right)+\frac{74}{5}.

$$y=10-3a$$

Substitute 10-3a for y and 8+5a for x in the equation z=2x+3y-46.

$$z=2\left(8+5a\right)+3\left(10-3a\right)-46$$

Calculate z from z=2\left(8+5a\right)+3\left(10-3a\right)-46.

$$z=a$$

The system is now solved.

$$x=8+5a y=10-3a z=a$$