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?

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.

## Comments

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.