Solve 1/3(A-V)+2/A+v=2-1/3v/v

$$\frac { \frac { 1 } { 3 } ( A - V ) + 2 } { A + v } = \frac { 2 - \frac { 1 } { 3 } v } { v }$$

Solve for V

$$V=v+2A-\frac{6A}{v}$$

$$A\neq -v\text{ and }v\neq 0$$

Steps for Solving Linear Equation

Multiply both sides of the equation by v\left(v+A\right), the least common multiple of A+v,v.

$$v\left(\frac{1}{3}\left(A-V\right)+2\right)=\left(v+A\right)\left(2-\frac{1}{3}v\right)$$

Use the distributive property to multiply \frac{1}{3} by A-V.

$$v\left(\frac{1}{3}A-\frac{1}{3}V+2\right)=\left(v+A\right)\left(2-\frac{1}{3}v\right)$$

Use the distributive property to multiply v by \frac{1}{3}A-\frac{1}{3}V+2.

$$\frac{1}{3}vA-\frac{1}{3}vV+2v=\left(v+A\right)\left(2-\frac{1}{3}v\right)$$

Use the distributive property to multiply v+A by 2-\frac{1}{3}v.

$$\frac{1}{3}vA-\frac{1}{3}vV+2v=2v-\frac{1}{3}v^{2}+2A-\frac{1}{3}Av$$

Subtract \frac{1}{3}vA from both sides.

$$-\frac{1}{3}vV+2v=2v-\frac{1}{3}v^{2}+2A-\frac{1}{3}Av-\frac{1}{3}vA$$

Combine -\frac{1}{3}Av and -\frac{1}{3}vA to get -\frac{2}{3}Av.

$$-\frac{1}{3}vV+2v=2v-\frac{1}{3}v^{2}+2A-\frac{2}{3}Av$$

Subtract 2v from both sides.

$$-\frac{1}{3}vV=2v-\frac{1}{3}v^{2}+2A-\frac{2}{3}Av-2v$$

Combine 2v and -2v to get 0.

$$-\frac{1}{3}vV=-\frac{1}{3}v^{2}+2A-\frac{2}{3}Av$$

The equation is in standard form.

$$\left(-\frac{v}{3}\right)V=-\frac{2Av}{3}-\frac{v^{2}}{3}+2A$$

Divide both sides by -\frac{1}{3}v.

$$\frac{\left(-\frac{v}{3}\right)V}{-\frac{v}{3}}=\frac{-\frac{2Av}{3}-\frac{v^{2}}{3}+2A}{-\frac{v}{3}}$$

Dividing by -\frac{1}{3}v undoes the multiplication by -\frac{1}{3}v.

$$V=\frac{-\frac{2Av}{3}-\frac{v^{2}}{3}+2A}{-\frac{v}{3}}$$

Divide -\frac{v^{2}}{3}+2A-\frac{2Av}{3} by -\frac{1}{3}v.

$$V=v+2A-\frac{6A}{v}$$

Solve for A

$$\left\{\begin{matrix}A=-\frac{v\left(v-V\right)}{2\left(v-3\right)}\text{, }&v\neq 0\text{ and }V\neq 6-v\text{ and }v\neq 3\\A\neq -3\text{, }&v=3\text{ and }V=3\end{matrix}\right.$$