Is it proper to manipulate the variables in a slope equation before substituting the value of the coordnates?

This question is actually specific to DataQuest mission 158-7 “Understanding Limits” at At the end of the mission, the limit of the slope equation was calculated by simplifying the numerator and eliminating equivalent terms. The two points in question were part of the graph for:


I assume f(x2) and f(x1) are both solvable thru that same equation.

To quote the mission:


My question is, can I not modify the slope equation itself and compute the SLOPE - not just the LIMIT of the SLOPE - of the two points as -3? Why can’t I just state the slope equation using the points as variables x and x+h then manipulate the variables and consider the non-indeterminate answer the slope?

I haven’t found a way myself, but is there a way of writing/arranging the slope equation so that I do not need to define the slope as a limit? Is defining the tangent lines slope really only possible thru limits?

Can you exemplify what you mean?

It already is in the DQ Courses category.

Short answer, yes. This particular use of limits is a bit of mathematical sleight of hand that allows us to calculate the slope of a line using only one point. Its complex and confusing but its pretty dang cool how it works.