Solve (x+y)dx+xdy=-7

$$( x + y ) d x + x d y = - 7$$

Solve for d

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

$$x\neq -2y\text{ and }x\neq 0$$

Steps for Solving Linear Equation

Use the distributive property to multiply x+y by d.

$$\left(xd+yd\right)x+xdy=-7$$

Use the distributive property to multiply xd+yd by x.

$$dx^{2}+ydx+xdy=-7$$

Combine ydx and xdy to get 2ydx.

$$dx^{2}+2ydx=-7$$

Combine all terms containing d.

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

The equation is in standard form.

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

Divide both sides by x^{2}+2xy.

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

Dividing by x^{2}+2xy undoes the multiplication by x^{2}+2xy.

$$d=-\frac{7}{x^{2}+2xy}$$

Divide -7 by x^{2}+2xy.

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

Solve for x (complex solution)

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

$$x=-\frac{\sqrt{\left(dy\right)^{2}-7d}}{d}-y\text{, }d\neq 0$$

Solve for x

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

$$x=-\frac{\sqrt{\left(dy\right)^{2}-7d}}{d}-y\text{, }\left(d\neq 0\text{ and }|y|\geq \frac{7}{\sqrt{7d}}\right)\text{ or }d<0$$