Hi
I'm stuck on this problem: How many positive integers (x,y) are solutions of the equation
(1/x) + (2/y) = (3/19).

I'm hoping that some could tell me how to approach these type of questions in general and what to
look out for. Also anyone got any good sites relating properties of shapes and numbers?

kryptos

What do you mean how many solutions? You mean how many zeros?

That means how many values of x and y are there that satisfy that equation.
(1/x) + (2/y) = (3/19). For example there may be 2 values of x and y that satisfy that equation.

I forgot to mention this, that that question was a multiple choice one:
So one of these is the answer: 0, 1, 2, 3 or more than 3.

This equation had an infinity of solutions !!!!

You can express it like that :
y = 38x / (3x-19)
which is the equation of an hyperbole...

x y
----------------
1 -2,375
2 -5,846153846
3 -11,4
4 -21,71428571
5 -47,5
6 -228
7 133
8 60,8
9 42,75
10 34,54545455
11 29,85714286
12 26,82352941
13 24,7
14 23,13043478
15 21,92307692
16 20,96551724
17 20,1875
18 19,54285714
19 19
20 18,53658537

I did mention positive integers. Simplifying the expression gives 3xy = 19y + 38x or y = 38x / (3x-19) but then what do u do after that??. I know the answer is 3 (wrote a simple program to vertify it, and its correct) but i don't know how to get the answer mathematically.

Hey there,
From y = 38*x/(3*x-19) => 3*y = 38*3*x/(3*x-19) = (38*(3*x-19)+38*19)/(3*x-19) = 38 + 38*19/(3*x-19)
Now we know (3*x-19) must be a factor of 38*19. Test +-1, +-2, +-19, +-38, +-361, +-722 you can get x and y.

Thanks man!

Sorry Kryptos, I focussed on your 2nd post where it was not anymore mentionned that this should be positive integers...

In addition to quangntenemy post, as you requested the positive integers, you don't have to test the negative values (-1, -2, -19, -38, -361, -722)
because you can see easily that before -19 'y' would be negative, for -19 'x' & 'y' would be equal to 0 which should be consider as neutral, and after
-19 'x' would be negative...