Solve {l}{2x=3y-3}{z=4x-5y+7}{text{Solvefor}atext{where}}{a=z}

$$\left. \begin{array} { l } { 2 x = 3 y - 3 }\\ { z = 4 x - 5 y + 7 }\\ { \text{Solve for } a \text{ where} } \\ { a = z } \end{array} \right.$$

Solve for x, y, z

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

$$y=a-1$$

$$z=a$$

Short Steps Using Substitution

Reorder the equations.

$$z=4x-5y+7 2x=3y-3 a=z$$

Substitute 4x-5y+7 for z in the equation a=z.

$$a=4x-5y+7$$

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

$$y=\frac{2}{3}x+1 x=-\frac{7}{4}+\frac{5}{4}y+\frac{1}{4}a$$

Substitute \frac{2}{3}x+1 for y in the equation x=-\frac{7}{4}+\frac{5}{4}y+\frac{1}{4}a.

$$x=-\frac{7}{4}+\frac{5}{4}\left(\frac{2}{3}x+1\right)+\frac{1}{4}a$$

Solve x=-\frac{7}{4}+\frac{5}{4}\left(\frac{2}{3}x+1\right)+\frac{1}{4}a for x.

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

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

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

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

$$y=-1+a$$

Substitute -1+a for y and -3+\frac{3}{2}a for x in the equation z=4x-5y+7.

$$z=4\left(-3+\frac{3}{2}a\right)-5\left(-1+a\right)+7$$

Calculate z from z=4\left(-3+\frac{3}{2}a\right)-5\left(-1+a\right)+7.

$$z=a$$

The system is now solved.

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