Solve x^3=ax+b-y^2

$$x ^ { 3 } = a x + b - y ^ { 2 }$$

Solve for a (complex solution)

$$\left\{\begin{matrix}a=\frac{x^{3}+y^{2}-b}{x}\text{, }&x\neq 0\\a\in \mathrm{C}\text{, }&b=y^{2}\text{ and }x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$ax+b-y^{2}=x^{3}$$

Subtract b from both sides.

$$ax-y^{2}=x^{3}-b$$

Add y^{2} to both sides.

$$ax=x^{3}-b+y^{2}$$

The equation is in standard form.

$$xa=x^{3}+y^{2}-b$$

Divide both sides by x.

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

Dividing by x undoes the multiplication by x.

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

Solve for a

$$\left\{\begin{matrix}a=\frac{x^{3}+y^{2}-b}{x}\text{, }&x\neq 0\\a\in \mathrm{R}\text{, }&x=0\text{ and }b=y^{2}\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$ax+b-y^{2}=x^{3}$$

Subtract b from both sides.

$$ax-y^{2}=x^{3}-b$$

Add y^{2} to both sides.

$$ax=x^{3}-b+y^{2}$$

The equation is in standard form.

$$xa=x^{3}+y^{2}-b$$

Divide both sides by x.

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

Dividing by x undoes the multiplication by x.

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

Solve for b

$$b=x^{3}-ax+y^{2}$$

Solution Steps

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

$$ax+b-y^{2}=x^{3}$$

Subtract ax from both sides.

$$b-y^{2}=x^{3}-ax$$

Add y^{2} to both sides.

$$b=x^{3}-ax+y^{2}$$