Solve sqrt{x}y=sqrt{y}

$$\sqrt { x } y = \sqrt { y }$$

Solve for y (complex solution)

$$\left\{\begin{matrix}\\y=0\text{, }&\text{unconditionally}\\y=\frac{1}{x}\text{, }&x\neq 0\end{matrix}\right.$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{x}y\right)^{2}=\left(\sqrt{y}\right)^{2}$$

Expand \left(\sqrt{x}y\right)^{2}.

$$\left(\sqrt{x}\right)^{2}y^{2}=\left(\sqrt{y}\right)^{2}$$

Calculate \sqrt{x} to the power of 2 and get x.

$$xy^{2}=\left(\sqrt{y}\right)^{2}$$

Calculate \sqrt{y} to the power of 2 and get y.

$$xy^{2}=y$$

Subtract y from both sides.

$$xy^{2}-y=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute x for a, -1 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-1\right)±\sqrt{1}}{2x}$$

Take the square root of 1.

$$y=\frac{-\left(-1\right)±1}{2x}$$

The opposite of -1 is 1.

$$y=\frac{1±1}{2x}$$

Now solve the equation y=\frac{1±1}{2x} when ± is plus. Add 1 to 1.

$$y=\frac{2}{2x}$$

Divide 2 by 2x.

$$y=\frac{1}{x}$$

Now solve the equation y=\frac{1±1}{2x} when ± is minus. Subtract 1 from 1.

$$y=\frac{0}{2x}$$

Divide 0 by 2x.

$$y=0$$

The equation is now solved.

$$y=\frac{1}{x} y=0$$

Substitute \frac{1}{x} for y in the equation \sqrt{x}y=\sqrt{y}.

$$\sqrt{x}\times \frac{1}{x}=\sqrt{\frac{1}{x}}$$

Simplify. The value y=\frac{1}{x} satisfies the equation.

$$x^{-\frac{1}{2}}=x^{-\frac{1}{2}}$$

Substitute 0 for y in the equation \sqrt{x}y=\sqrt{y}.

$$\sqrt{x}\times 0=\sqrt{0}$$

Simplify. The value y=0 satisfies the equation.

$$0=0$$

List all solutions of \sqrt{x}y=\sqrt{y}.

$$y=\frac{1}{x} y=0$$

Solve for x

$$\left\{\begin{matrix}x=\frac{1}{y}\text{, }&y>0\\x\geq 0\text{, }&y=0\end{matrix}\right.$$

Steps for Solving Radical Equations

Divide both sides by y.

$$\frac{y\sqrt{x}}{y}=\frac{\sqrt{y}}{y}$$

Dividing by y undoes the multiplication by y.

$$\sqrt{x}=\frac{\sqrt{y}}{y}$$

Divide \sqrt{y} by y.

$$\sqrt{x}=\frac{1}{\sqrt{y}}$$

Square both sides of the equation.

$$x=\frac{1}{y}$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=\frac{1}{y}\text{, }&y\neq 0\\x\in \mathrm{C}\text{, }&y=0\end{matrix}\right.$$

Solve for y

$$\left\{\begin{matrix}y=0\text{, }&x\geq 0\\y=\frac{1}{x}\text{, }&x>0\end{matrix}\right.$$