Solve frac{x-{x}_{1}}{y-f{x}_{1}}=frac{{x}_{2}-{x}_{1}}{f{x}_{2}-f{x}_{1}}

$$\frac{ x- { x }_{ 1 } }{ y-f { x }_{ 1 } } = \frac{ { x }_{ 2 } - { x }_{ 1 } }{ f { x }_{ 2 } -f { x }_{ 1 } }$$

Solve for x

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

$$y\neq fx_{1}\text{ and }x_{1}\neq x_{2}\text{ and }f\neq 0$$

Steps for Solving Linear Equation

Multiply both sides of the equation by f\left(-x_{1}+x_{2}\right)\left(fx_{1}-y\right), the least common multiple of y-fx_{1},fx_{2}-fx_{1}.

$$\left(fx_{1}-fx_{2}\right)\left(x-x_{1}\right)=\left(fx_{1}-y\right)\left(x_{2}-x_{1}\right)$$

Use the distributive property to multiply fx_{1}-fx_{2} by x-x_{1}.

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

Use the distributive property to multiply fx_{1}-y by x_{2}-x_{1}.

$$fx_{1}x-fx_{1}^{2}-fx_{2}x+fx_{2}x_{1}=fx_{1}x_{2}-fx_{1}^{2}-yx_{2}+yx_{1}$$

Add fx_{1}^{2} to both sides.

$$fx_{1}x-fx_{2}x+fx_{2}x_{1}=fx_{1}x_{2}-fx_{1}^{2}-yx_{2}+yx_{1}+fx_{1}^{2}$$

Combine -fx_{1}^{2} and fx_{1}^{2} to get 0.

$$fx_{1}x-fx_{2}x+fx_{2}x_{1}=fx_{1}x_{2}-yx_{2}+yx_{1}$$

Subtract fx_{2}x_{1} from both sides.

$$fx_{1}x-fx_{2}x=fx_{1}x_{2}-yx_{2}+yx_{1}-fx_{2}x_{1}$$

Combine fx_{1}x_{2} and -fx_{2}x_{1} to get 0.

$$fx_{1}x-fx_{2}x=-yx_{2}+yx_{1}$$

Combine all terms containing x.

$$\left(fx_{1}-fx_{2}\right)x=-yx_{2}+yx_{1}$$

The equation is in standard form.

$$\left(fx_{1}-fx_{2}\right)x=x_{1}y-x_{2}y$$

Divide both sides by fx_{1}-fx_{2}.

$$\frac{\left(fx_{1}-fx_{2}\right)x}{fx_{1}-fx_{2}}=\frac{y\left(x_{1}-x_{2}\right)}{fx_{1}-fx_{2}}$$

Dividing by fx_{1}-fx_{2} undoes the multiplication by fx_{1}-fx_{2}.

$$x=\frac{y\left(x_{1}-x_{2}\right)}{fx_{1}-fx_{2}}$$

Divide y\left(-x_{2}+x_{1}\right) by fx_{1}-fx_{2}.

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

Solve for f

$$\left\{\begin{matrix}f=\frac{y}{x}\text{, }&x\neq x_{1}\text{ and }y\neq 0\text{ and }x\neq 0\text{ and }x_{1}\neq x_{2}\\f\neq 0\text{, }&x_{1}\neq 0\text{ and }y=0\text{ and }x=0\text{ and }x_{1}\neq x_{2}\end{matrix}\right.$$