Hi,

Let x represent  and y the other

the difference of their reciprocals is 3

1/y - (1/x)= 3 

Multiplying thru with xy to clear the demominators

x - y  = 3xy

the sum of two numbers is the same as their product****

x + y = xy

Adding the equation together & eliminating the y variable

2x = 4xy

1/2 = y

substituting for y 

x + 1/2 = x/2

x/2 = -1/2

x = -1

Checking our answer

 2 - (-1) = 3 

Answer by Edwin McCravy(20056)   (Show Source): You can put this solution on YOUR website!
 the sum of two numbers is the same as their product, and the difference of thier recoprocals is 3. find the numbers

Let the numbers be x and y

The sum of the the numbers = x + y

The product of the numbers = xy

The reciprocal of x is 

The reciprocal of y is 

"the sum of two numbers x+y is the same as their product xy",

so  

 

"the difference of their recoprocals  is 3",

so  



So we have the system of equations 



Clear the second equation of fractions by multiplying through by LCD xy







So the system  of equations is now



Write the left side of the first equation in reverse order:



Adding the two equations term by term,

2y = 4xy

Get 0 on the right side:

2y - 4xy = 0

Divide through by 2

y - 2xy = 0

y(1 - 2x) = 0

Set each factor = 0

y = 0;   1 - 2x = 0

            -2x = -1

              x = 

y = 0 must be discarded because the reciprocal of 0 is not defined.

So x =   

Substituting in 


 


Multiplying through by LCD of 2

1 + 2y = y

     y = -1

So the numbers are  and 

Checking:

The sum of the the numbers = 

The product of the numbers = 

That checks because they are both the same.

The reciprocal of  is 2

The reciprocal of -1 is  or -1.

The difference of these reciprocals is 

So that checks.

Edwin
 

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