   SEARCH HOME Math Central Quandaries & Queries  Mary, a student: can you help, 2/35= 1/x +1/y ; x Hi Mary,

I think there maybe an error in your given question. As it is written now, there are no two real numbers that would satisfy the solution.

2/35=y/(xy) + x /(xy)

2/35 = (y+x)/(xy)
so 2 = y+x & 35=xy. Since 2-y=x, we can substitute into 35=xy

35=(2-y)y

35=2y-y2

y2-2y+35=0

Using the quadratic formula would result in answers of x = 1-i√34 & y=1+i√34, two complex solutions.

If the equation were 2/35=1/x-1/y, our solution would have two real solutions. If you find a common denominator and simplify the right side of the equation:
2/35=y/(xy) - x /(xy)

2/35 = (y-x)/(xy)

so 2 = y-x & 35=xy. At this point you can either use substitution to find your values of x & y or think "what two numbers subtract to 2 and multiply to 35?"
The numbers 5 & 7 work.

Hope this helps,

Janice & Gary

Mary wrote back

Thank you for your help,
I may have found another solution. PLEASE let me know if this works.

2/35= 6/105= 1/x +1/y x<y

6/105= 5/105 + 1/105

6/105= 1/21 + 1/105 x<y 21<105

or 2/35= 8/140= 1/x +1/y x<y

8/140= 7/140 + 1/140

8/140= 1/20 + 1/140 x<y 20<140

Mary,

Your answer is correct, our answer is incorrect. We overlooked the fact that a fracion can be written in many equivalent ways. In particular for your first solution you wrote 2/35 as 6/105 and then solved the problem.

Thank you for point out our error.

Harley     Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.