# Derivation of linear interpolation?

Anyone know a good derivation of the linear interpolation:

$$\frac{y-y_0}{x-x_0}=\frac{y_1-y_0}{x_1-x_0}$$

Wikipedia gives one, which I don't understand.

Since it is a linear interpolation, just consider a straight line $y=a+ bx$ which goes through two points $(x_0,y_0)$ and $(x_1,y_1)$. So $$y_0=a+b x_0$$ $$y_1=a+b x_1$$ Solve for $a,b$.

For any two points: $$(x_0 ,y_0 )$$ and $$(x_1 ,y_1 )$$

$$y_0 =mx_0 +b$$ and $$y_1 =mx_1 +b$$

Solve for $$b$$ for each equation...

Then set the two equal to each other...

Next solve for $$m$$

you get: $$m=(y_0 -y_1 )/(x_0 -x_1 )$$

Since this is true for any two points on the line, its also true for the first point and another point on the line where you may know the $$y$$ but not the $$x$$: say $$(x,y_2 )$$

so then: $$m= \frac{y_0 -y_1 }{x_0 -x_1 }=\frac{y_0 -y_2 }{x_0 -x}.$$

There it is!

• It's helpful if downvoters can explain their action. As someone who is trying to understand this question, I have no way of knowing what is wrong with this answer. Jun 2, 2021 at 18:43