Question 216492
For this problem let number 1 be X and number 2 be Y


{{{X+Y=9}}}
{{{X^2+Y^2=41}}}

There are many ways to solve this... but lets try the most simple way.
Find the Square numbers up to 8

{{{1^2=1}}}
{{{2^2=4}}}
{{{3^2=9}}}
{{{4^2=16}}}
{{{5^2=25}}}
{{{6^2=36}}}
{{{7^2=49}}}
{{{8^2=64}}}

Right off the bat you can eliminate the last two...  
This leaves us with...

1
4
9
16
25
36

The easy way to do it from here is pretend each of those are X... W stands for whole number.  Plug in the X's... If 41-X does not equal a whole number, ELIMINATE it.  

{{{41-X=W^2}}}

Let's try...

{{{41-1=40}}}

40 is not a square number... ELIMINATED  a

Next number...

{{{41-4=37}}}

37 is not a square number either... 2 is eliminated

{{{41-9=32}}}

32 is not a square number... and since 3+6=9, 6 is eliminated as well...

This leaves us with 4 and 5 left... let's try..

{{{41-16=25}}}
{{{16=4^2}}}
{{{25=5^2}}}

The two numbers are 4 and 5.