help desk - slope formula

Saxon has failed me.

What is the answer to this question:

Write a short explanation of why the slope formula works, and include an explanation of why either point can be point 1 or point 2.

The solution manual simply refers you back to the Lesson, which doesn't "explain" why the slope formula works.

I said that the slope formula "works" because slope is "defined" as change in y coordinate divided by change in x coordinate.

However, I can't explain why it doesn't matter which point you designate as point 1. The best I could come up with was something along the lines of...."subtraction is defined as addition of the opposite".... and I'm not even sure that's relevant, to tell the truth.


from Tracy:

I assume the formula you are using for the slope of a line is

(y1-y2)/(x1-x2)

It doesn't matter whether you chose point 1 or point 2 because if the slope is positive and you chose a lower point as point 1 and do the subtractions you wind up with a negative number on top and a negative number below the divsor line, and a negative number divided by a negative number is a positive number.

If the slope is negative, then you wind up with one positive number and one negative number regardless of what you pick as point 1 so the result of the divison is a negative number.

from Greta

When you switch the two points, the sign in each difference changes but the absolute value remains the same: X1 - X2 is the opposite of X2 - X1, and Y1 - Y2 is the opposite of Y2 - Y1. The absolute value of the quotient (the difference in Y divided by the difference in X) will not change since the absolute values of the two differences don't change. The sign of the quotient will not change either: If you had two positive differences before, you will end up with two negative differences after switching the points; either way, the quotient is positive. (Same result if you started with two negative differences.) If you had one negative and one positive difference before (negative quotient), each sign will switch, and you will still have one negative and one positive difference, resulting in a negative quotient.

from Steve H:

(y1-y2)/(x1-x2)

= -(y2-y1)/[-(x2-x1)]

= -1*(y2-y1)/[-1*(x2-x1)]

= -1/-1 * (y2-y1)/(x2-x1)

= (y2-y1)/(x2-x1)

I really appreciate this.

I "know" why you can use either point as "Point 1," but I have no idea how to express it....which leads me to my next plea for help.