Solve left(2x-2right)dx+(3y+7)y=0

$$\left( 2x-2 \right) dx+(3y+7)y=0$$

Solve for d (complex solution)

$$\left\{\begin{matrix}d=-\frac{y\left(3y+7\right)}{2x\left(x-1\right)}\text{, }&x\neq 1\text{ and }x\neq 0\\d\in \mathrm{C}\text{, }&\left(y=-\frac{7}{3}\text{ and }x=0\right)\text{ or }\left(y=-\frac{7}{3}\text{ and }x=1\right)\text{ or }\left(y=0\text{ and }x=0\right)\text{ or }\left(y=0\text{ and }x=1\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$\left(2xd-2d\right)x+\left(3y+7\right)y=0$$

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

$$2dx^{2}-2dx+\left(3y+7\right)y=0$$

Use the distributive property to multiply 3y+7 by y.

$$2dx^{2}-2dx+3y^{2}+7y=0$$

Subtract 3y^{2} from both sides. Anything subtracted from zero gives its negation.

$$2dx^{2}-2dx+7y=-3y^{2}$$

Subtract 7y from both sides.

$$2dx^{2}-2dx=-3y^{2}-7y$$

Combine all terms containing d.

$$\left(2x^{2}-2x\right)d=-3y^{2}-7y$$

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

$$\frac{\left(2x^{2}-2x\right)d}{2x^{2}-2x}=-\frac{y\left(3y+7\right)}{2x^{2}-2x}$$

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

$$d=-\frac{y\left(3y+7\right)}{2x^{2}-2x}$$

Divide -y\left(7+3y\right) by 2x^{2}-2x.

$$d=-\frac{y\left(3y+7\right)}{2x\left(x-1\right)}$$

Solve for d

$$\left\{\begin{matrix}d=-\frac{y\left(3y+7\right)}{2x\left(x-1\right)}\text{, }&x\neq 1\text{ and }x\neq 0\\d\in \mathrm{R}\text{, }&\left(y=-\frac{7}{3}\text{ and }x=0\right)\text{ or }\left(y=-\frac{7}{3}\text{ and }x=1\right)\text{ or }\left(y=0\text{ and }x=0\right)\text{ or }\left(y=0\text{ and }x=1\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$\left(2xd-2d\right)x+\left(3y+7\right)y=0$$

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

$$2dx^{2}-2dx+\left(3y+7\right)y=0$$

Use the distributive property to multiply 3y+7 by y.

$$2dx^{2}-2dx+3y^{2}+7y=0$$

Subtract 3y^{2} from both sides. Anything subtracted from zero gives its negation.

$$2dx^{2}-2dx+7y=-3y^{2}$$

Subtract 7y from both sides.

$$2dx^{2}-2dx=-3y^{2}-7y$$

Combine all terms containing d.

$$\left(2x^{2}-2x\right)d=-3y^{2}-7y$$

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

$$\frac{\left(2x^{2}-2x\right)d}{2x^{2}-2x}=-\frac{y\left(3y+7\right)}{2x^{2}-2x}$$

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

$$d=-\frac{y\left(3y+7\right)}{2x^{2}-2x}$$

Divide -y\left(7+3y\right) by 2x^{2}-2x.

$$d=-\frac{y\left(3y+7\right)}{2x\left(x-1\right)}$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=\frac{\sqrt{d\left(d-14y-6y^{2}\right)}+d}{2d}\text{; }x=\frac{-\sqrt{d\left(d-14y-6y^{2}\right)}+d}{2d}\text{, }&d\neq 0\\x\in \mathrm{C}\text{, }&\left(y=-\frac{7}{3}\text{ or }y=0\right)\text{ and }d=0\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=\frac{\sqrt{d\left(d-14y-6y^{2}\right)}+d}{2d}\text{; }x=\frac{-\sqrt{d\left(d-14y-6y^{2}\right)}+d}{2d}\text{, }&\left(y\neq 0\text{ and }y\neq -\frac{7}{3}\text{ and }d=6y^{2}+14y\right)\text{ or }\left(d\leq 6y^{2}+14y\text{ and }d<0\right)\text{ or }\left(d\geq 6y^{2}+14y\text{ and }d>0\right)\\x\in \mathrm{R}\text{, }&\left(y=-\frac{7}{3}\text{ or }y=0\right)\text{ and }d=0\end{matrix}\right.$$