Solve {l}{2x-y=1}{z=-5x+3y}{text{Solvefor}atext{where}}{a=z}

$$\left. \begin{array} { l } { 2 x - y = 1 }\\ { z = -5 x + 3 y }\\ { \text{Solve for } a \text{ where} } \\ { a = z } \end{array} \right.$$

Solve for x, y, z

$$x=a+3$$

$$y=2a+5$$

$$z=a$$

Short Steps Using Substitution

Reorder the equations.

$$z=-5x+3y 2x-y=1 a=z$$

Substitute -5x+3y for z in the equation a=z.

$$a=-5x+3y$$

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

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

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

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

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

$$x=3+a$$

Substitute 3+a for x in the equation y=2x-1.

$$y=2\left(3+a\right)-1$$

Calculate y from y=2\left(3+a\right)-1.

$$y=5+2a$$

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

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

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

$$z=a$$

The system is now solved.

$$x=3+a y=5+2a z=a$$