Help with two variable equation

Question

I need help with this equation which was given to my son who is in 4rth grade.

Solve in positive integers the equation: $xy=10(x+y)$.

score 3 · Answer 1 · answered Oct 10 '13 at 18:34

3

HINT:

$$(x-m)(y-m)=xy-m(x+y)+m^2$$

answered Oct 10 '13 at 18:34

lab bhattacharjee

274,582

ya beat me to it – what'sup Oct 10 '13 at 18:35

score 2 · Accepted Answer · answered Oct 10 '13 at 18:56

2

$xy =10x+10y$ $\implies$ $y=\frac {10x}{x-10}$.
Now use hit and trial or you can easily see that y has an integral value when x equals an integral value of 10 except 1.

answered Oct 10 '13 at 18:56

This is the correct answer... I was almost to write it, but Shrayansh was faster :) – Cristi Oct 10 '13 at 18:57
Could you give me a more detailed explanation? How do you obtain the solution (35,14) for example?/ – Anton Oct 10 '13 at 22:39
The choice of x is taken such that $x-10$ is a factor of $10x$. If we choose $ x=35 $ $\implies$ $x-10=25$ and $10x=35*10$ which is a factor of $25$ and thus y is an integer $ y=14 $. – Oct 11 '13 at 03:25

score 1 · Answer 3 · answered Oct 10 '13 at 18:47

1

To follow up on lab's excellent hint, that means that $(x-10)(y-10)=100$, so you can have $(x,y)=(20,20),(35,14),(60,12),(30,15),(110,11)$ (and also swapping $x,y$). A 4th grader might find one of these by trial and error.

answered Oct 10 '13 at 18:47

vadim123

82,796

Help with two variable equation

3 Answers3