fitting a curve to a straight line

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In my opinion the easiest way to do the fit is to use the find_fit function, which is standard in SAGE. For example:

sage: x = [1, 3, 4]   
sage: y = [1.2, 4.5, 3.6]
sage: var('a,b,t')
sage: model(t)=a*t+b
sage: data=zip(x,y)
sage: fit=find_fit(data,model,solution_dict=True) # Dictionary is optional but handy
sage: model.subs(fit) # Show the model
t |--> 0.92142857142840007*t + 0.64285714285636364
sage: plot(model.subs(fit),(t,0,5))+points(data,size=20,color='red') # Nice plot

When you have a huge number of points and you want just a polynomial fit, I found that it is (numerically) better to use the polyfit function from numpy:

sage: import numpy as np
sage: a,b=np.polyfit(x,y,1)
sage: a,b
(0.92142857142857137, 0.64285714285714401)

I have a feeling that the answer to this ask.sagemath question may also be helpful. It mentions the built-in find_fit method.

I didn't find any reference to linear regression in the SAGE reference manual. That means you have to use Maxima directly. Fortunately that's not so difficult.

It's easier if you present the data as a list of pairs (x,y). Try this in a SAGE notebook cell:

maxima.load("stats")
x=maxima.simple_linear_regression([[0,1],[1,2],[2,4]],conf_level=0.95)

x

You are presented a whole bunch of statistics about the regression. From there on I guess you can manage on your own. Good luck.