Solve 1/y=1/2x^2+1

$$\frac{ 1 }{ y } = \frac{ 1 }{ 2 { x }^{ 2 } } +1$$

Solve for y (complex solution)

$$y=\frac{2x^{2}}{2x^{2}+1}$$

$$x\neq 0\text{ and }x\neq -\frac{\sqrt{2}i}{2}\text{ and }x\neq \frac{\sqrt{2}i}{2}$$

Steps for Solving Linear Equation

Variable y cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 2yx^{2}, the least common multiple of y,2x^{2}.

$$2x^{2}=y+2yx^{2}$$

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

$$y+2yx^{2}=2x^{2}$$

Combine all terms containing y.

$$\left(1+2x^{2}\right)y=2x^{2}$$

The equation is in standard form.

$$\left(2x^{2}+1\right)y=2x^{2}$$

Divide both sides by 2x^{2}+1.

$$\frac{\left(2x^{2}+1\right)y}{2x^{2}+1}=\frac{2x^{2}}{2x^{2}+1}$$

Dividing by 2x^{2}+1 undoes the multiplication by 2x^{2}+1.

$$y=\frac{2x^{2}}{2x^{2}+1}$$

Variable y cannot be equal to 0.

$$y=\frac{2x^{2}}{2x^{2}+1}\text{, }y\neq 0$$

Solve for y

$$y=\frac{2x^{2}}{2x^{2}+1}$$

$$x\neq 0$$

Steps for Solving Linear Equation

Variable y cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 2yx^{2}, the least common multiple of y,2x^{2}.

$$2x^{2}=y+2yx^{2}$$

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

$$y+2yx^{2}=2x^{2}$$

Combine all terms containing y.

$$\left(1+2x^{2}\right)y=2x^{2}$$

The equation is in standard form.

$$\left(2x^{2}+1\right)y=2x^{2}$$

Divide both sides by 1+2x^{2}.

$$\frac{\left(2x^{2}+1\right)y}{2x^{2}+1}=\frac{2x^{2}}{2x^{2}+1}$$

Dividing by 1+2x^{2} undoes the multiplication by 1+2x^{2}.

$$y=\frac{2x^{2}}{2x^{2}+1}$$

Variable y cannot be equal to 0.

$$y=\frac{2x^{2}}{2x^{2}+1}\text{, }y\neq 0$$

Solve for x (complex solution)

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

$$x=\frac{i\left(y-1\right)^{-\frac{1}{2}}\sqrt{2y}}{2}\text{, }y\neq 0\text{ and }y\neq 1$$

Solve for x

$$x=\frac{\sqrt{-\frac{2y}{y-1}}}{2}$$

$$x=-\frac{\sqrt{-\frac{2y}{y-1}}}{2}\text{, }y>0\text{ and }y<1$$