Solve {l}{4x+3y=-6}{z=3x-2y}{text{Solvefor}atext{where}}{a=z}

$$\left. \begin{array} { l } { 4 x + 3 y = -6 }\\ { z = 3 x - 2 y }\\ { \text{Solve for } a \text{ where} } \\ { a = z } \end{array} \right.$$

Solve for x, y, z

$$x=\frac{3a-12}{17}$$

$$y=\frac{-4a-18}{17}$$

$$z=a$$

Short Steps Using Substitution

Reorder the equations.

$$z=3x-2y 4x+3y=-6 a=z$$

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

$$a=3x-2y$$

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

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

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

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

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

$$x=-\frac{12}{17}+\frac{3}{17}a$$

Substitute -\frac{12}{17}+\frac{3}{17}a for x in the equation y=-2-\frac{4}{3}x.

$$y=-2-\frac{4}{3}\left(-\frac{12}{17}+\frac{3}{17}a\right)$$

Calculate y from y=-2-\frac{4}{3}\left(-\frac{12}{17}+\frac{3}{17}a\right).

$$y=-\frac{18}{17}-\frac{4}{17}a$$

Substitute -\frac{18}{17}-\frac{4}{17}a for y and -\frac{12}{17}+\frac{3}{17}a for x in the equation z=3x-2y.

$$z=3\left(-\frac{12}{17}+\frac{3}{17}a\right)-2\left(-\frac{18}{17}-\frac{4}{17}a\right)$$

Calculate z from z=3\left(-\frac{12}{17}+\frac{3}{17}a\right)-2\left(-\frac{18}{17}-\frac{4}{17}a\right).

$$z=a$$

The system is now solved.

$$x=-\frac{12}{17}+\frac{3}{17}a y=-\frac{18}{17}-\frac{4}{17}a z=a$$