My older son is doing a little Algebra review and he told me that he was having trouble understanding “standard form” for lines.
Since he’s doing this review mostly on his own, I have to confess that I did not remember what the “standard form” for lines even was! We talked through some basics about lines and some different ways to represent the equation of a line for our project tonight:
Here are some of the basics about lines:
and here’s our conversation about some of the different ways to write the equation of a line:
I enjoyed talking about these ideas with him and definitely learned why different ways to write equations of lines could be confusing. In part, it is hard to understand that x and y are variables and the other letters in the equations are constants. Another source of confusion, I think, is probably similar to the confusion about why 0.999999 repeating and 1 are the “same” number. How can these two representations which seem totally different mean the same thing?
Mike's Math Page 
Fun trick. Say you want the line through (x1,y1) and (x2,y2).
Make the 3×3 matrix
x y 1
x1 y1 1
x2 y2 1
(variables in the first row, numbers in the other rows).
Its determinant will be x*(something) + y*(something) + 1*something, so if you set that to zero, it’s the equation of some line.
If (x,y) = (x1,y1) or (x2,y2), then there’s a repeated row, so det=0. I.e. the line does indeed go through the two points.