Solve (2x-1)dx+63y+9dy=0

$$( 2 x - 1 ) d x + 63 y + 9 d y = 0$$

Solve for d (complex solution)

$$\left\{\begin{matrix}d=-\frac{63y}{2x^{2}-x+9y}\text{, }&y\neq -\frac{x\left(2x-1\right)}{9}\\d\in \mathrm{C}\text{, }&\left(x=\frac{1}{2}\text{ or }x=0\right)\text{ and }y=0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$\left(2xd-d\right)x+63y+9dy=0$$

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

$$2dx^{2}-dx+63y+9dy=0$$

Subtract 63y from both sides. Anything subtracted from zero gives its negation.

$$2dx^{2}-dx+9dy=-63y$$

Combine all terms containing d.

$$\left(2x^{2}-x+9y\right)d=-63y$$

Divide both sides by 2x^{2}-x+9y.

$$\frac{\left(2x^{2}-x+9y\right)d}{2x^{2}-x+9y}=-\frac{63y}{2x^{2}-x+9y}$$

Dividing by 2x^{2}-x+9y undoes the multiplication by 2x^{2}-x+9y.

$$d=-\frac{63y}{2x^{2}-x+9y}$$

Solve for d

$$\left\{\begin{matrix}d=-\frac{63y}{2x^{2}-x+9y}\text{, }&y\neq -\frac{x\left(2x-1\right)}{9}\\d\in \mathrm{R}\text{, }&\left(x=\frac{1}{2}\text{ or }x=0\right)\text{ and }y=0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$\left(2xd-d\right)x+63y+9dy=0$$

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

$$2dx^{2}-dx+63y+9dy=0$$

Subtract 63y from both sides. Anything subtracted from zero gives its negation.

$$2dx^{2}-dx+9dy=-63y$$

Combine all terms containing d.

$$\left(2x^{2}-x+9y\right)d=-63y$$

Divide both sides by 2x^{2}-x+9y.

$$\frac{\left(2x^{2}-x+9y\right)d}{2x^{2}-x+9y}=-\frac{63y}{2x^{2}-x+9y}$$

Dividing by 2x^{2}-x+9y undoes the multiplication by 2x^{2}-x+9y.

$$d=-\frac{63y}{2x^{2}-x+9y}$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=\frac{\sqrt{d\left(d-504y-72dy\right)}+d}{4d}\text{; }x=\frac{-\sqrt{d\left(d-504y-72dy\right)}+d}{4d}\text{, }&d\neq 0\\x\in \mathrm{C}\text{, }&y=0\text{ and }d=0\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=\frac{\sqrt{d\left(d-504y-72dy\right)}+d}{4d}\text{; }x=\frac{-\sqrt{d\left(d-504y-72dy\right)}+d}{4d}\text{, }&\left(y<\frac{1}{72}\text{ and }d\geq \frac{504y}{1-72y}\text{ and }d>0\right)\text{ or }\left(d\leq \frac{504y}{1-72y}\text{ and }y>\frac{1}{72}\text{ and }d>0\right)\text{ or }\left(d\geq \frac{504y}{1-72y}\text{ and }y\geq \frac{1}{72}\text{ and }d<0\right)\text{ or }\left(y=\frac{1}{72}\text{ and }d<0\right)\text{ or }\left(y\leq \frac{1}{72}\text{ and }d\leq \frac{504y}{1-72y}\text{ and }d<0\right)\text{ or }\left(y\neq 0\text{ and }d=\frac{504y}{1-72y}\text{ and }y\neq \frac{1}{72}\right)\\x\in \mathrm{R}\text{, }&y=0\text{ and }d=0\end{matrix}\right.$$