Question 358617: the sum of two numbers is the same as their product, and the difference of thier reciprocals is 3. find the numbers Found 2 solutions by ewatrrr, Edwin McCravy:Answer by ewatrrr(24785) (Show Source):
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
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
