I'm reading a book on math and in the first chapter, in a section discussing existence and uniqueness with regard to polynomials, the author presents a formula that looks like this (and although it might be guessed, what this is supposed to illustrate about polynomials and uniqueness isn't important):
[MATH]f(x) = \frac{y_1}{x_1 - x_2}(x - x_2) + \frac{y_2}{x_2 - x_1}(x - x_1)[/MATH]
He follows this immediately with "and simplify with typical algebra to get":
[MATH]f(x) = \frac{x_1 y_2 - x_2 y_1}{x_1 - x_2} + (\frac{y_1 - y_2}{x_1 - x_2})x[/MATH]
Then adding (in order to get on with the point of the discussion there): "Instead of doing all that algebra I..."
Which has me flabbergasted, because I consider myself reasonably well versed in algebra (objective testing backs that up); I even have a small stack of algebra I and II texts and workbooks which at least a couple of times a week I spend a few hours with, cycling through the material to keep it familiar. I've been doing that for a few years, completing one of them at least several times, and there's nothing in those other texts I have been so utterly stumped by -- in fact, as far as I can remember, this is the first time I've ever gone online to look for help.
But when I look at these formulas, I do not see how the second one is a product of "simplifying with typical algebra". I don't see any way to get from one to the other. This resembles it, and is a such a simplification, but I cannot get from there to the second one either:
[MATH]f(x) = \frac{y_1 x - y_1 x_2}{x_1 - x_2} + \frac{y_2 x - y_2 x_1}{x_2 -x_1}[/MATH]
[MATH]f(x) = \frac{y_1}{x_1 - x_2}(x - x_2) + \frac{y_2}{x_2 - x_1}(x - x_1)[/MATH]
He follows this immediately with "and simplify with typical algebra to get":
[MATH]f(x) = \frac{x_1 y_2 - x_2 y_1}{x_1 - x_2} + (\frac{y_1 - y_2}{x_1 - x_2})x[/MATH]
Then adding (in order to get on with the point of the discussion there): "Instead of doing all that algebra I..."
Which has me flabbergasted, because I consider myself reasonably well versed in algebra (objective testing backs that up); I even have a small stack of algebra I and II texts and workbooks which at least a couple of times a week I spend a few hours with, cycling through the material to keep it familiar. I've been doing that for a few years, completing one of them at least several times, and there's nothing in those other texts I have been so utterly stumped by -- in fact, as far as I can remember, this is the first time I've ever gone online to look for help.
But when I look at these formulas, I do not see how the second one is a product of "simplifying with typical algebra". I don't see any way to get from one to the other. This resembles it, and is a such a simplification, but I cannot get from there to the second one either:
[MATH]f(x) = \frac{y_1 x - y_1 x_2}{x_1 - x_2} + \frac{y_2 x - y_2 x_1}{x_2 -x_1}[/MATH]