Solve w*f(x)=x^2+3x/x^2-1

$$w \cdot f ( x ) = \frac { x ^ { 2 } + 3 x } { x ^ { 2 } - 1 }$$

Solve for f (complex solution)

$$\left\{\begin{matrix}f=\frac{x+3}{w\left(x^{2}-1\right)}\text{, }&x\neq -1\text{ and }x\neq 1\text{ and }w\neq 0\\f\in \mathrm{C}\text{, }&x=0\text{ or }\left(x=-3\text{ and }w=0\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by \left(x-1\right)\left(x+1\right).

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

Use the distributive property to multiply wfx by x-1.

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

Use the distributive property to multiply wfx^{2}-wfx by x+1 and combine like terms.

$$wfx^{3}-wfx=x^{2}+3x$$

Combine all terms containing f.

$$\left(wx^{3}-wx\right)f=x^{2}+3x$$

Divide both sides by wx^{3}-wx.

$$\frac{\left(wx^{3}-wx\right)f}{wx^{3}-wx}=\frac{x\left(x+3\right)}{wx^{3}-wx}$$

Dividing by wx^{3}-wx undoes the multiplication by wx^{3}-wx.

$$f=\frac{x\left(x+3\right)}{wx^{3}-wx}$$

Divide x\left(3+x\right) by wx^{3}-wx.

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

Solve for w (complex solution)

$$\left\{\begin{matrix}w=\frac{x+3}{f\left(x^{2}-1\right)}\text{, }&x\neq -1\text{ and }x\neq 1\text{ and }f\neq 0\\w\in \mathrm{C}\text{, }&x=0\text{ or }\left(x=-3\text{ and }f=0\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by \left(x-1\right)\left(x+1\right).

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

Use the distributive property to multiply wfx by x-1.

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

Use the distributive property to multiply wfx^{2}-wfx by x+1 and combine like terms.

$$wfx^{3}-wfx=x^{2}+3x$$

Combine all terms containing w.

$$\left(fx^{3}-fx\right)w=x^{2}+3x$$

Divide both sides by fx^{3}-fx.

$$\frac{\left(fx^{3}-fx\right)w}{fx^{3}-fx}=\frac{x\left(x+3\right)}{fx^{3}-fx}$$

Dividing by fx^{3}-fx undoes the multiplication by fx^{3}-fx.

$$w=\frac{x\left(x+3\right)}{fx^{3}-fx}$$

Divide x\left(3+x\right) by fx^{3}-fx.

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

Solve for f

$$\left\{\begin{matrix}f=\frac{x+3}{w\left(x^{2}-1\right)}\text{, }&w\neq 0\text{ and }|x|\neq 1\\f\in \mathrm{R}\text{, }&x=0\text{ or }\left(x=-3\text{ and }w=0\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by \left(x-1\right)\left(x+1\right).

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

Use the distributive property to multiply wfx by x-1.

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

Use the distributive property to multiply wfx^{2}-wfx by x+1 and combine like terms.

$$wfx^{3}-wfx=x^{2}+3x$$

Combine all terms containing f.

$$\left(wx^{3}-wx\right)f=x^{2}+3x$$

Divide both sides by wx^{3}-wx.

$$\frac{\left(wx^{3}-wx\right)f}{wx^{3}-wx}=\frac{x\left(x+3\right)}{wx^{3}-wx}$$

Dividing by wx^{3}-wx undoes the multiplication by wx^{3}-wx.

$$f=\frac{x\left(x+3\right)}{wx^{3}-wx}$$

Divide x\left(3+x\right) by wx^{3}-wx.

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

Solve for w

$$\left\{\begin{matrix}w=\frac{x+3}{f\left(x^{2}-1\right)}\text{, }&f\neq 0\text{ and }|x|\neq 1\\w\in \mathrm{R}\text{, }&x=0\text{ or }\left(x=-3\text{ and }f=0\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by \left(x-1\right)\left(x+1\right).

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

Use the distributive property to multiply wfx by x-1.

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

Use the distributive property to multiply wfx^{2}-wfx by x+1 and combine like terms.

$$wfx^{3}-wfx=x^{2}+3x$$

Combine all terms containing w.

$$\left(fx^{3}-fx\right)w=x^{2}+3x$$

Divide both sides by fx^{3}-fx.

$$\frac{\left(fx^{3}-fx\right)w}{fx^{3}-fx}=\frac{x\left(x+3\right)}{fx^{3}-fx}$$

Dividing by fx^{3}-fx undoes the multiplication by fx^{3}-fx.

$$w=\frac{x\left(x+3\right)}{fx^{3}-fx}$$

Divide x\left(3+x\right) by fx^{3}-fx.

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