SOLUTION: Hello! Could you give e chanse to have the solution of this problem? The different of two naural number is 9 and he different of their reciprocal is 9/190. Find the numbers. This

Algebra ->  Systems-of-equations -> SOLUTION: Hello! Could you give e chanse to have the solution of this problem? The different of two naural number is 9 and he different of their reciprocal is 9/190. Find the numbers. This      Log On


   

Question 202020: Hello! Could you give e chanse to have the solution of this problem?
The different of two naural number is 9 and he different of their reciprocal is 9/190. Find the numbers.
This was my way which doesn't go on:
x%2B9=y
x%2B9%2F190=1%2Fy
x%2B9%2F190=1%2F%28x%2B9%29
%28190x%2B9%29%28x%2B9%29=190
190x%5E2%2B1719x-109=0
x+=+%28-1719%2B-+sqrt%28+1719%5E2+%2B4%2A190%2A%28-109%29+%29%29%2F%282%2A190%29
or
x+=+%28-1719%2B-+sqrt%28+3037801+%29+%29%2F%282%2A190%29+
Since sqrt( 3037801 ) is not an integer number (it is a so called irrational number, not reducible to fractions like m/n), further reduction of this expression will not give you an integer result.
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
"The difference of two natural numbers is 9" translates to x-y=9

"the difference of their reciprocals is 9/190" means that 1%2Fy-1%2Fx=9%2F190


Note: if x%3Ey, then 1%2Fx%3C1%2Fy


x-y=9 Start with the first equation.


x=9%2By Add "y" to both sides.


1%2Fy-1%2Fx=9%2F190 Move onto the second equation


1%2Fy-1%2F%289%2By%29=9%2F190 Plug in x=9%2By


Multiply EVERY term by the LCD 190y%289%2By%29 to clear out the fractions.


190%289%2By%29-190y=9y%289%2By%29 Simplify


1710%2B190y-190y=81y%2B9y%5E2 Distribute


1710%2B190y-190y-81y-9y%5E2=0 Get all terms to the left side.


-9y%5E2-81y%2B1710=0 Combine like terms.


Notice that the quadratic -9y%5E2-81y%2B1710 is in the form of Ay%5E2%2BBy%2BC where A=-9, B=-81, and C=1710


Let's use the quadratic formula to solve for "y":


y+=+%28-B+%2B-+sqrt%28+B%5E2-4AC+%29%29%2F%282A%29 Start with the quadratic formula


y+=+%28-%28-81%29+%2B-+sqrt%28+%28-81%29%5E2-4%28-9%29%281710%29+%29%29%2F%282%28-9%29%29 Plug in A=-9, B=-81, and C=1710


y+=+%2881+%2B-+sqrt%28+%28-81%29%5E2-4%28-9%29%281710%29+%29%29%2F%282%28-9%29%29 Negate -81 to get 81.


y+=+%2881+%2B-+sqrt%28+6561-4%28-9%29%281710%29+%29%29%2F%282%28-9%29%29 Square -81 to get 6561.


y+=+%2881+%2B-+sqrt%28+6561--61560+%29%29%2F%282%28-9%29%29 Multiply 4%28-9%29%281710%29 to get -61560


y+=+%2881+%2B-+sqrt%28+6561%2B61560+%29%29%2F%282%28-9%29%29 Rewrite sqrt%286561--61560%29 as sqrt%286561%2B61560%29


y+=+%2881+%2B-+sqrt%28+68121+%29%29%2F%282%28-9%29%29 Add 6561 to 61560 to get 68121


y+=+%2881+%2B-+sqrt%28+68121+%29%29%2F%28-18%29 Multiply 2 and -9 to get -18.


y+=+%2881+%2B-+261%29%2F%28-18%29 Take the square root of 68121 to get 261.


y+=+%2881+%2B+261%29%2F%28-18%29 or y+=+%2881+-+261%29%2F%28-18%29 Break up the expression.


y+=+%28342%29%2F%28-18%29 or y+=++%28-180%29%2F%28-18%29 Combine like terms.


y+=+-19 or y+=+10 Simplify.


So the possible solutions for "y" are y+=+-19 or y+=+10

However, the problem stated that the numbers are natural. So we have to throw out y+=+-19


So the only answer for "y" is y=10. This means that "x" is x=9%2By=9%2B10=19


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Answer:


So the solutions are x=19 and y=10


So the two numbers are 19 and 10


Take note that 19-10=9 and 1%2F10-1%2F19=19%2F190-10%2F190=%2819-10%29%2F190=9%2F190