Solve y^-1=x^3+1/2

$${ y }^{ -1 } = \frac{ { x }^{ 3 } +1 }{ 2 }$$

Solve for y (complex solution)

$$y=\frac{2}{x^{3}+1}$$

$$x\neq -1\text{ and }x\neq \frac{-\sqrt{3}i+1}{2}\text{ and }x\neq \frac{1+\sqrt{3}i}{2}$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 2.

$$2y^{-1}=x^{3}+1$$

Reorder the terms.

$$2\times \frac{1}{y}=x^{3}+1$$

Variable y cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by y.

$$2\times 1=yx^{3}+y$$

Multiply 2 and 1 to get 2.

$$2=yx^{3}+y$$

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

$$yx^{3}+y=2$$

Combine all terms containing y.

$$\left(x^{3}+1\right)y=2$$

Divide both sides by x^{3}+1.

$$\frac{\left(x^{3}+1\right)y}{x^{3}+1}=\frac{2}{x^{3}+1}$$

Dividing by x^{3}+1 undoes the multiplication by x^{3}+1.

$$y=\frac{2}{x^{3}+1}$$

Divide 2 by x^{3}+1.

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

Variable y cannot be equal to 0.

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

Solve for y

$$y=\frac{2}{x^{3}+1}$$

$$x\neq -1$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 2.

$$2y^{-1}=x^{3}+1$$

Reorder the terms.

$$2\times \frac{1}{y}=x^{3}+1$$

Variable y cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by y.

$$2\times 1=yx^{3}+y$$

Multiply 2 and 1 to get 2.

$$2=yx^{3}+y$$

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

$$yx^{3}+y=2$$

Combine all terms containing y.

$$\left(x^{3}+1\right)y=2$$

Divide both sides by x^{3}+1.

$$\frac{\left(x^{3}+1\right)y}{x^{3}+1}=\frac{2}{x^{3}+1}$$

Dividing by x^{3}+1 undoes the multiplication by x^{3}+1.

$$y=\frac{2}{x^{3}+1}$$

Divide 2 by x^{3}+1.

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

Variable y cannot be equal to 0.

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

Solve for x (complex solution)

$$x=e^{\frac{2\pi i}{3}}\sqrt[3]{-1+\frac{2}{y}}$$

$$x=\sqrt[3]{-1+\frac{2}{y}}$$

$$x=e^{\frac{4\pi i}{3}}\sqrt[3]{-1+\frac{2}{y}}\text{, }y\neq 0$$

Solve for x

$$x=\sqrt[3]{-1+\frac{2}{y}}$$

$$y\neq 0$$