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, y=167.

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.
 
https://web.williams.edu/Mathematics/sjmiller/public_html/BrownClasses/54/handouts/MethodLeastSquares.pdf

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