Solve 4Axy+2y^2=-10x

$$4 A x y + 2 y ^ { 2 } = - 10 x$$

Solve for A (complex solution)

$$\left\{\begin{matrix}A=\frac{-5x-y^{2}}{2xy}\text{, }&y\neq 0\text{ and }x\neq 0\\A\in \mathrm{C}\text{, }&x=0\text{ and }y=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract 2y^{2} from both sides.

$$4Axy=-10x-2y^{2}$$

The equation is in standard form.

$$4xyA=-10x-2y^{2}$$

Divide both sides by 4xy.

$$\frac{4xyA}{4xy}=\frac{-10x-2y^{2}}{4xy}$$

Dividing by 4xy undoes the multiplication by 4xy.

$$A=\frac{-10x-2y^{2}}{4xy}$$

Divide -2y^{2}-10x by 4xy.

$$A=-\frac{y}{2x}-\frac{5}{2y}$$

Solve for A

$$\left\{\begin{matrix}A=\frac{-5x-y^{2}}{2xy}\text{, }&y\neq 0\text{ and }x\neq 0\\A\in \mathrm{R}\text{, }&x=0\text{ and }y=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract 2y^{2} from both sides.

$$4Axy=-10x-2y^{2}$$

The equation is in standard form.

$$4xyA=-10x-2y^{2}$$

Divide both sides by 4xy.

$$\frac{4xyA}{4xy}=\frac{-10x-2y^{2}}{4xy}$$

Dividing by 4xy undoes the multiplication by 4xy.

$$A=\frac{-10x-2y^{2}}{4xy}$$

Divide -10x-2y^{2} by 4xy.

$$A=-\frac{y}{2x}-\frac{5}{2y}$$

Solve for x

$$x=-\frac{y^{2}}{2Ay+5}$$

$$y=0\text{ or }A\neq -\frac{5}{2y}$$

Steps for Solving Linear Equation

Add 10x to both sides.

$$4Axy+2y^{2}+10x=0$$

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

$$4Axy+10x=-2y^{2}$$

Combine all terms containing x.

$$\left(4Ay+10\right)x=-2y^{2}$$

Divide both sides by 4Ay+10.

$$\frac{\left(4Ay+10\right)x}{4Ay+10}=-\frac{2y^{2}}{4Ay+10}$$

Dividing by 4Ay+10 undoes the multiplication by 4Ay+10.

$$x=-\frac{2y^{2}}{4Ay+10}$$

Divide -2y^{2} by 4Ay+10.

$$x=-\frac{y^{2}}{2Ay+5}$$