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E182631 · Mar 25, 2016, 11:58 PM

In the equations below, A through G are all distinct positive integers. Solve for X and Y.9925 = A² + B² = C² + D²E = C + D - 128F = (B + D) - (A + C)G = C - DX = 14 - 12D - 10C + 63AY = 7D - 35C + 26B - 2 (This is not homework. It's a geocaching puzzle. I'm not sure how to approach it, and am seeking assistance. Thanks in advance for any suggestions.)

smoothy · Mar 26, 2016, 04:24 AM

Originally Posted by E182631

In the equations below, A through G are all distinct positive integers. Solve for X and Y.9925 = A² + B² = C² + D²E = C + D - 128F = (B + D) - (A + C)G = C - DX = 14 - 12D - 10C + 63AY = 7D - 35C + 26B - 2

Since this is your homework and not ours... show us your work and what you got for an answer.

E182631 · Mar 26, 2016, 06:52 PM

Originally Posted by smoothy

Since this is your homework and not ours... show us your work and what you got for an answer.

It's not homework. It's a geocaching puzzle, and I wasn't sure how to approach it. I don't expect it to be solved, but some suggestion about how to go about it would be appreciated. Thanks in advance to those willing to offer a constructive response.

ebaines · Mar 28, 2016, 08:55 AM

There seems to be an error in what you wrote. This part: C^2+ D^2E= C + D - 128F can't be true if C, D, E and F are all positive integers. The reason is that C^2 > C and D^2 > D for all positive values of C and D >1 (and we know that that C and D are not both equal to 1). Hence the left side of the equation is greater than the right side assuming E>0 and F>0. Please check that the problem as you wrote it is precisely as it was given to you.

E182631 · Mar 28, 2016, 04:18 PM

Perhaps I didn't space things out correctly when I typed the question initially since I had a limited amount of space to type. I'll post them in sections.

In the equations below, A through G are all distinct positive integers. Solve for X and Y.

9925 = A² + B² = C² + D²

^{E = C + D - 128

F = (B + D) - (A + C)

G = C - D}

X = 14 - 12D - 10C + 63AY = 7D - 35C + 26B - 2

This is exactly what the problem is showing me. Thank you for your help.

ebaines · Mar 29, 2016, 05:51 AM

This is better, although I do find there are two possible solutions.

I'll give you a hint to get started. First, notice the problem states that unknowns A through G are all positive and distinct integers. Also note that the unknowns A, B, C and D are the really important ones - you need to know their values in order to determine X and Y. The unknowns E, F, and G only serve to help you figure out the values of A through D by requiring that they are positive integers. So to get started you need to come up with possible values for A. B, C and D, based on the first two equations:

$A^2 + B^2 = 9925$

and

$C^2 + D^2 = 9925$

It turns out that there are only three possible combinations of integers whose squares add to 9925. One of those pairs must be (A,B), another is (C,D), and the third pair is not used. You must determine these three pairs of integers and try them with the equations for E, F and G and see which combination(s) give you positive distinct integers. For example the equation $E=C+D-128$ limits C and D to being only one pair of the three possible pairs of integers - otherwise E would end up being a negative number. And from $G = C-D$ you know that C is the larger value of the (C,D) pair. So now you know the values of the (C, D) pair, and can figure out which is the
(A, B) pair by trial and error (there are only 4 possible combinations to try). So - go ahead and get started on finding the three possible pairs that satisfy the first two equations, and post back if you are still having difficulty.