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

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

Solve for x, y, z

$$x=a-5$$

$$y=15-2a$$

$$z=a$$

Short Steps Using Substitution

Reorder the equations.

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

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

$$a=3x+y$$

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

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

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

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

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

$$x=-5+a$$

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

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

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

$$y=15-2a$$

Substitute 15-2a for y and -5+a for x in the equation z=3x+y.

$$z=3\left(-5+a\right)+15-2a$$

Calculate z from z=3\left(-5+a\right)+15-2a.

$$z=a$$

The system is now solved.

$$x=-5+a y=15-2a z=a$$