Me and my friend are revising for the [UK] STEP exams, and came across this problem:
Given:
p² + 4q² + 9r² = 729 and 8p + 8q + 3r = 243
Solve p, q, and r.
Now, this question is actually the 3rd part of one question; so I was convinced that we had to make use of one of our previous answers for that question, in order to solve this one (as is usually the case on STEP papers).
However, my friend used the fact that 243 was a factor of 729, and solved it as so:
3 * (8p + 8q + 3r) = 3 * 243
24p + 24q + 9r = 729
Seeing as p² + 4q² + 9r² also equals 729, then:
p² + 4q² + 9r² = 24p + 24q + 9r
We then move everything from the RHS to the LHS:
p² - 24p + 4q² - 24q + 9r² - 9r = 0
Now factor out p, q, and r:
p(p - 24) + q(4q - 24) + r(9r - 9) = 0
Now this is where I had a problem. My mate seemed to think that in this case, each term must each equal 0 -- despite my teacher and I both saying that you can't assume that.
However, we had a bit of a surprise when he done it anyway...
The p=q=r=0 solution does not work, but p=24, q=6, and r=1 is exactly what me and my teacher achieved when we used a [longer] way which *did* refer back to one of our previous questions.
My teacher (who is actually very good at maths) could not see if there was any sort of logic as to how and why this worked; his first thoughts were that it was coincidence, but perhaps it was intentional?
Can anyone see whether or not there's some logic behind why it worked?
I'd re-read the question. Does it say "find a solution" or "find the solution" or something misleading? There is no unique solution as both WaterStreet and ozo have said.
WaterStreet,
Welcome to the only other page with an & in it (I think). :-)
Paul
0
Solve your biggest tech problems alongside global tech experts with 1:1 help.
And then he would have had certainty that his solution is correct (or at least one of the correct solutions). Then he only had to establish that this system has only 1 real solution to be sure that his solution is the one and only correct one.
So, to summarise : it's NOT a coincidence that he arrived at the solution this way, BUT he should have been more complete ...
>>There are an infinite number of solutions ... - ozo
Correct, ozo. The standard way is to introduce a lambda multiplier and multiply and add the two equations together (the example uses lambda=1) and then say that, because lambda is arbitary, the terms must go out one by one. Hence the solution found is when lambda is 1.