<snip> >>> >>>But your solution has more than that because it gives >>> >>>y = (5f_1 - 3f_2 + 21g_1 - g_2)/4 >>> >>>as a solution as well. > > > [Rick Decker] > >>No. However, it would be interesting to see how you got this.
<snip enough so that this subthread is completely unreadable to anyone who hasn't been following closely> > > Are you psychic or what?
Yes, and I knew you'd ask that.
> I had no idea how he came up with the thing > containing 21g_1, and never would have guessed he was just pulling it out of > his butt :-)
Surely you're not surprised. > > >>Let h_1 and h_2 be chosen so that h_1 * h_2 = 21 * T >> >> h_1 + h_2 = 10*y + 42*z + 19*f_1 - 3*f_2  >> h_2 - h_1 = 4*y - 5*f_1 + 3*f_2  > > > I think you meant to write h_1 - h_2 on the LHS of .
Indeed I did. > > >>Then we can write >> >> (10*y + 42*z + 19*f_1 - 3*f_2)^2 = (4*y - 5*f_1 + 3*f_2)^2 + 84*T >> >>in the form >> >> (h_1 + h_2)^2 = (h_1 - h_2)^2 + 4 * h_1 * h_2 > > > This part would be clearer with the correction above.
Yes. > > >>Then, from  and  we solve for y to get >> >> y = (5*f_1 - 3*f_2 + h_1 - h_2) / 4  > > > While this conclusion _needs_ the correction above.
Yes, yes. > > >>Then, since h_1 * h_2 = 21 * T = 21 * g_1 * g_2 we may as well >>pick h_1 = 21 * g_1 and h_2 = g_2 so  becomes >> >> y = (5*f_1 - 3*f_2 + 21*g_1 - g_2) / 4 >> >>Right? > > > Yes, you are psychic!
I knew you'd say that. > > >>If that was your reasoning, it's wrong. You can't pick any old >>values for h_1 and h_2. Watch: >> >>Solving  and  for h_1 and h_2 we get >> >> h_1 = 7(y + 3 * z + f_1) >> h_2 = 3(y + 7 * z + 4 * f_1 - f_2) > > > That also needs the correction above ;-)
(Grr). Yes, yes, yes! > > >>But from your original four linear equations we can derive >> >> g_1 = y + 3 * z + f_1 >> g_2 = y + 7 * z + 4 * f_1 - f_2 >> >>in other words, we are forced to choose >> >> h_1 = 7 * g_1 >> h_2 = 3 * g_2 >> >>and not your h_1 = 21 * g_1 and h_2 = g_2. > > > And to force the conclusion, in that case  becomes > > y = (5*f_1 - 3*f_2 + 7*g_1 - 3*g_2)/4 > > > But let's give James something else to worry about :-) Take > > (42*z + 10*y - 3*f_2 + 19*f_1)^2 = (4*y + 3*f_2 - 5*f_1)^2 + 84*T > > expand it, use the quadratic equation to solve for y, and then substitute to > get rid of z and T: > > z = -(3*f_1 - f_2 + g_1 - g_2)/4 > T = g_1*g_2 > > The result is: > > y = (5*f_1 - 3*f_2 + 5*g_1 - 5*g_2 +/- 2*(g_1 + g_2))/4 > > Pick "+" and you get the result James wants: > > y = (5*f_1 - 3*f_2 + 7*g_1 - 3*g_2)/4 > > Pick "-" and it's different: > > y = (5*f_1 - 3*f_2 + 3*g_1 - 7*g_2)/4 > > Woo hoo! Centuries of mathematics down the tubes again, or can James spot > the bogosity? Hint #1: this isn't an algebraic error; you really do get > that result for y. Hint #2: you get the same two results for y if you do > the same thing but starting from > > (2*y + 10*z + 5*f_1 - f_2)^2 = (4*z + 3*f_1 - f_2)^2 + 4*T > > instead.
Hehe. I predict that this section (cute, BTW) will generate no response. > > >><snip>
>>>I wonder if you just lied. > > >>You just can't resist, can you? Are you naturally boorish, or do >>you have to work at it? > > > I strongly suspect that bit of gratuitous assholishness was deliberate. God > only knows why, but James got it into his head that he needs to _provoke_ > people into replying when he thinks they know something he wants to find > out. That's just his despicable way of trying to goad you into doing his > work for him. It's especially idiotic in this case, since if he had any > memory he'd recall that you typically respond much better to polite requests > than to his stupid baiting tactics. > > But, in this case, I'm afraid what he'll take away is "ha! it worked again", > without a shadow of a clue that it was neither necessary nor helpful to > behave like an ass. > Sadly, I predict you're right again.