How do you model a linear equation when there's no apparent pattern between x and y? For example x=30, y=134, x=31, y=155, x=33, y=165, x=35,...

Hello!

It is obvious that there is no linear formula exactly connecting these x's and y's, if we consider the slopes between neighbor points:

`(155 - 134)/(31 - 30) = 21,` `(165 - 155)/(33 - 31) = 5,` `(167 - 165)/(35 - 33) = 1.`

For a single line, all these slopes must be the same.

But this isn't the whole story. We may seek such a line `y=ax+b` that would be the closest to all these points. The simplest criteria of such a proximity is the least squares one, which means we try to minimize

`sum_(k=1)^n (y_n-(ax_n+b))^2.`

This problem has the exact unique answer (see for example the link attached). We have to compute the numbers

`p=sum_(k=1)^n x_k^2,`  `q=sum_(k=1)^n x_k,`  `r=sum_(k=1)^n x_k y_k`  and  `s=sum_(k=1)^n y_k.`

In our case  `p=4175,` `q=129,` `r=20115` and `s=621.` Then we solve the linear system for the unknowns a and b,

`pa+qb=r,`  `qa+nb=s`  (here n=4).

I hope you know how to solve such systems, the solution for this is `a=351/59,` `b=-2160/59.`