Solve y=sqrt{p}x-2qy+y

$$y = \sqrt { p } x - 2 q y + y$$

Solve for q (complex solution)

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

Steps for Solving Linear Equation

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

$$\sqrt{p}x-2qy+y=y$$

Subtract \sqrt{p}x from both sides.

$$-2qy+y=y-\sqrt{p}x$$

Subtract y from both sides.

$$-2qy=y-\sqrt{p}x-y$$

Combine y and -y to get 0.

$$-2qy=-\sqrt{p}x$$

The equation is in standard form.

$$\left(-2y\right)q=-\sqrt{p}x$$

Divide both sides by -2y.

$$\frac{\left(-2y\right)q}{-2y}=-\frac{\sqrt{p}x}{-2y}$$

Dividing by -2y undoes the multiplication by -2y.

$$q=-\frac{\sqrt{p}x}{-2y}$$

Divide -\sqrt{p}x by -2y.

$$q=\frac{\sqrt{p}x}{2y}$$

Solve for p

$$\left\{\begin{matrix}p=4\times \left(\frac{qy}{x}\right)^{2}\text{, }&\left(q\leq 0\text{ or }x<0\text{ or }y\geq 0\right)\text{ and }\left(y\leq 0\text{ or }x<0\text{ or }q\geq 0\right)\text{ and }\left(q\leq 0\text{ or }y\leq 0\text{ or }x>0\right)\text{ and }x\neq 0\text{ and }\left(x>0\text{ or }q\geq 0\text{ or }y\geq 0\right)\\p\geq 0\text{, }&\left(q=0\text{ and }x=0\right)\text{ or }\left(y=0\text{ and }x=0\right)\end{matrix}\right.$$

Steps for Solving Radical Equations

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

$$\sqrt{p}x-2qy+y=y$$

Add 2qy to both sides.

$$\sqrt{p}x+y=y+2qy$$

Subtract y from both sides.

$$\sqrt{p}x=y+2qy-y$$

Combine y and -y to get 0.

$$\sqrt{p}x=2qy$$

Divide both sides by x.

$$\frac{x\sqrt{p}}{x}=\frac{2qy}{x}$$

Dividing by x undoes the multiplication by x.

$$\sqrt{p}=\frac{2qy}{x}$$

Square both sides of the equation.

$$p=\frac{4q^{2}y^{2}}{x^{2}}$$

Solve for q

$$\left\{\begin{matrix}q=\frac{\sqrt{p}x}{2y}\text{, }&y\neq 0\text{ and }p\geq 0\\q\in \mathrm{R}\text{, }&\left(p=0\text{ or }x=0\right)\text{ and }p\geq 0\text{ and }y=0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$\sqrt{p}x-2qy+y=y$$

Subtract \sqrt{p}x from both sides.

$$-2qy+y=y-\sqrt{p}x$$

Subtract y from both sides.

$$-2qy=y-\sqrt{p}x-y$$

Combine y and -y to get 0.

$$-2qy=-\sqrt{p}x$$

The equation is in standard form.

$$\left(-2y\right)q=-\sqrt{p}x$$

Divide both sides by -2y.

$$\frac{\left(-2y\right)q}{-2y}=-\frac{\sqrt{p}x}{-2y}$$

Dividing by -2y undoes the multiplication by -2y.

$$q=-\frac{\sqrt{p}x}{-2y}$$

Divide -\sqrt{p}x by -2y.

$$q=\frac{\sqrt{p}x}{2y}$$

Solve for p (complex solution)

$$\left\{\begin{matrix}p=4\times \left(\frac{qy}{x}\right)^{2}\text{, }&x\neq 0\text{ and }\left(|arg(x\sqrt{\left(\frac{qy}{x}\right)^{2}})-arg(qy)|<\pi \text{ or }q=0\text{ or }y=0\right)\\p\in \mathrm{C}\text{, }&\left(q=0\text{ or }y=0\right)\text{ and }x=0\end{matrix}\right.$$