Solve dx=2v+3/3v+4dv

$$d x = \frac { 2 v + 3 } { 3 v + 4 } d v$$

Solve for d

$$\left\{\begin{matrix}d=0\text{, }&v\neq -\frac{4}{3}\\d\in \mathrm{R}\text{, }&x=\frac{v\left(2v+3\right)}{3v+4}\text{ and }v\neq -\frac{4}{3}\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 3v+4.

$$dx\left(3v+4\right)=\left(2v+3\right)dv$$

Use the distributive property to multiply dx by 3v+4.

$$3dxv+4dx=\left(2v+3\right)dv$$

Use the distributive property to multiply 2v+3 by d.

$$3dxv+4dx=\left(2vd+3d\right)v$$

Use the distributive property to multiply 2vd+3d by v.

$$3dxv+4dx=2dv^{2}+3dv$$

Subtract 2dv^{2} from both sides.

$$3dxv+4dx-2dv^{2}=3dv$$

Subtract 3dv from both sides.

$$3dxv+4dx-2dv^{2}-3dv=0$$

Combine all terms containing d.

$$\left(3xv+4x-2v^{2}-3v\right)d=0$$

The equation is in standard form.

$$\left(3vx+4x-2v^{2}-3v\right)d=0$$

Divide 0 by 3xv+4x-2v^{2}-3v.

$$d=0$$

Solve for v

$$\left\{\begin{matrix}\\v=\frac{\sqrt{9x^{2}+14x+9}+3x-3}{4}\text{; }v=\frac{-\sqrt{9x^{2}+14x+9}+3x-3}{4}\text{, }&\text{unconditionally}\\v\neq -\frac{4}{3}\text{, }&d=0\end{matrix}\right.$$