Find the solution where x and y are integers to:
1/x + 1/y = 1/14
I can use a computer to get a list of solutions:
7, -14
10, -35
12, -84
13, -182
15, -210
16, 112
18, 63
21, 42
28, 28
but I'm having trouble solving this analytically and coming to a general formula.
1/x + 1/y = 1/14
(x+y)/xy=1/14 Inverse
xy/(x+y)=14
xy=14x+14y
xy-14x=14y
x(y-14)=14y
xy=14x+14y
xy-14y=14x
y(x-14)=14x Divide with (x-14)
y=14x/(x-14)
Solutions:
x-14#0
and
y=14x/(x-14) x#0
Do not forget that, since x and y are symmetric, they are interchangeable on your list.
Example: if (13, -182) works, then (-182, 13) works as well.
Go to:
wolframalpha com
When page be open in rectangle type:
1/x + 1/y = 1/14
and click option =
When you see results couple times click option: More Solutions
bosnian, i already used wolfram alpha to get the solutions I listed. I want to derive the solution analytically.
anon, I believe you just did algebraic rearrangement.
To solve the equation 1/x + 1/y = 1/14, we can use a technique called "Simon's Favorite Factoring Trick." This trick helps us rewrite an equation in a way that allows us to factor it easily.
Starting with the equation 1/x + 1/y = 1/14, we can cross-multiply to eliminate the fractions:
y + x = xy/14
Now, let's rearrange the equation to bring all terms to one side:
xy - 14x - 14y = 0
To make this equation easier to work with, we can add 196 to both sides of the equation:
xy - 14x - 14y + 196 = 196
Now, notice that the left side of the equation can be factored using Simon's Favorite Factoring Trick:
(xy - 14x - 14y + 196) = (x - 14)(y - 14) = 196
We want to find integer solutions for x and y, so we need to find pairs of integers whose product is equal to 196.
To do this, we can find the factors of 196, which are:
1, 2, 4, 7, 14, 28, 49, 98, 196
Now, since (x - 14)(y - 14) = 196, we can set each factor pair equal to the corresponding expression:
x - 14 = 1, and y - 14 = 196
x - 14 = 2, and y - 14 = 98
x - 14 = 4, and y - 14 = 49
x - 14 = 7, and y - 14 = 28
x - 14 = 14, and y - 14 = 14
...
Solving each equation, we obtain the following values for x and y:
x = 15, y = 210
x = 16, y = 112
x = 18, y = 84
x = 21, y = 42
x = 28, y = 28
These are the solutions in which x and y are integers that satisfy the equation 1/x + 1/y = 1/14.