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

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

Solve for x, y, z

$$x=\frac{15a}{11}$$

$$y=-\frac{17a}{11}$$

$$z=a$$

Short Steps Using Substitution

Multiply each equation by the least common multiple of denominators in it. Simplify.

$$20x+15y=3x z=3x+2y a=z$$

Reorder the equations.

$$z=3x+2y 20x+15y=3x a=z$$

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.

$$y=-\frac{17}{15}x x=\frac{1}{3}a-\frac{2}{3}y$$

Substitute -\frac{17}{15}x for y in the equation x=\frac{1}{3}a-\frac{2}{3}y.

$$x=\frac{1}{3}a-\frac{2}{3}\left(-\frac{17}{15}\right)x$$

Solve x=\frac{1}{3}a-\frac{2}{3}\left(-\frac{17}{15}\right)x for x.

$$x=\frac{15}{11}a$$

Substitute \frac{15}{11}a for x in the equation y=-\frac{17}{15}x.

$$y=-\frac{17}{15}\times \frac{15}{11}a$$

Calculate y from y=-\frac{17}{15}\times \frac{15}{11}a.

$$y=-\frac{17}{11}a$$

Substitute -\frac{17}{11}a for y and \frac{15}{11}a for x in the equation z=3x+2y.

$$z=3\times \frac{15}{11}a+2\left(-\frac{17}{11}\right)a$$

Calculate z from z=3\times \frac{15}{11}a+2\left(-\frac{17}{11}\right)a.

$$z=a$$

The system is now solved.

$$x=\frac{15}{11}a y=-\frac{17}{11}a z=a$$