Solve y=(2x-1)dx

$$y = ( 2 x - 1 ) d x$$

Solve for d (complex solution)

$$\left\{\begin{matrix}d=\frac{y}{x\left(2x-1\right)}\text{, }&x\neq \frac{1}{2}\text{ and }x\neq 0\\d\in \mathrm{C}\text{, }&\left(x=0\text{ or }x=\frac{1}{2}\right)\text{ and }y=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply 2x-1 by d.

$$y=\left(2xd-d\right)x$$

Use the distributive property to multiply 2xd-d by x.

$$y=2dx^{2}-dx$$

Swap sides so that all variable terms are on the left hand side.

$$2dx^{2}-dx=y$$

Combine all terms containing d.

$$\left(2x^{2}-x\right)d=y$$

Divide both sides by 2x^{2}-x.

$$\frac{\left(2x^{2}-x\right)d}{2x^{2}-x}=\frac{y}{2x^{2}-x}$$

Dividing by 2x^{2}-x undoes the multiplication by 2x^{2}-x.

$$d=\frac{y}{2x^{2}-x}$$

Divide y by 2x^{2}-x.

$$d=\frac{y}{x\left(2x-1\right)}$$

Solve for d

$$\left\{\begin{matrix}d=\frac{y}{x\left(2x-1\right)}\text{, }&x\neq \frac{1}{2}\text{ and }x\neq 0\\d\in \mathrm{R}\text{, }&\left(x=0\text{ or }x=\frac{1}{2}\right)\text{ and }y=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply 2x-1 by d.

$$y=\left(2xd-d\right)x$$

Use the distributive property to multiply 2xd-d by x.

$$y=2dx^{2}-dx$$

Swap sides so that all variable terms are on the left hand side.

$$2dx^{2}-dx=y$$

Combine all terms containing d.

$$\left(2x^{2}-x\right)d=y$$

Divide both sides by 2x^{2}-x.

$$\frac{\left(2x^{2}-x\right)d}{2x^{2}-x}=\frac{y}{2x^{2}-x}$$

Dividing by 2x^{2}-x undoes the multiplication by 2x^{2}-x.

$$d=\frac{y}{2x^{2}-x}$$

Divide y by 2x^{2}-x.

$$d=\frac{y}{x\left(2x-1\right)}$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=\frac{\sqrt{d\left(8y+d\right)}+d}{4d}\text{; }x=\frac{-\sqrt{d\left(8y+d\right)}+d}{4d}\text{, }&d\neq 0\\x\in \mathrm{C}\text{, }&y=0\text{ and }d=0\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=\frac{\sqrt{d\left(8y+d\right)}+d}{4d}\text{; }x=\frac{-\sqrt{d\left(8y+d\right)}+d}{4d}\text{, }&\left(y\leq -\frac{d}{8}\text{ and }d<0\right)\text{ or }\left(y\geq -\frac{d}{8}\text{ and }d>0\right)\text{ or }\left(d\neq 0\text{ and }y=-\frac{d}{8}\right)\\x\in \mathrm{R}\text{, }&y=0\text{ and }d=0\end{matrix}\right.$$