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

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

Solve for x, y, z

$$x=a-1$$

$$y=2-a$$

$$z=a$$

Short Steps Using Substitution

Reorder the equations.

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

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

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

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

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

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

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

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

$$x=-1+a$$

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

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

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

$$y=2-a$$

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

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

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

$$z=a$$

The system is now solved.

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