SOLUTION: The sum of the reciprocal of two numbers is 5. The product of the reciprocal of the two numbers and 3 is equal to 18. Find the two numbers

Algebra ->  Systems-of-equations -> SOLUTION: The sum of the reciprocal of two numbers is 5. The product of the reciprocal of the two numbers and 3 is equal to 18. Find the two numbers      Log On


   

Question 1081841: The sum of the reciprocal of two numbers is 5. The product of the reciprocal of the two
numbers and 3 is equal to 18. Find the two numbers
Found 2 solutions by josgarithmetic, ikleyn:
Answer by josgarithmetic(39613) About Me  (Show Source):
You can put this solution on YOUR website!
system%281%2Fx%2B1%2Fy=5%2C3%281%2Fx%29%281%2Fy%29=18%29

y%2F%28xy%29%2Bx%2F%28xy%29=5

%28x%2By%29%2F%28xy%29=5
-

3%2F%28xy%29=18
1%2F%28xy%29=6
-
%28x%2By%29%286%29=5
6x%2B6y=5

xy=1%2F6
y=1%2F%286x%29
-
6x%2B6%281%2F6x%29=5
6x%2B1%2Fx=5
6x%5E2%2B1=5x
6x%5E2-5x%2B1=0
-
x=%285%2B-+sqrt%2825-4%2A6%2A1%29%29%2F12
x=%285%2B-+1%29%2F12
highlight_green%28system%28x=1%2F3%2Cor%2Cx=1%2F2%29%29


Finding corresponding y
xy=1%2F6
y=1%2F%286x%29
y=1%2F%286%281%2F3%29%29
y=1%2F2
or
y=1%2F%286%281%2F2%29%29
y=1%2F3


The two numbers are highlight%28system%281%2F3%2Cand%2C1%2F2%29%29.

Answer by ikleyn(52754) About Me  (Show Source):
You can put this solution on YOUR website!
.
Let x and y be the two numbers under the question.

Then their reciprocals are 1%2Fx and 1%2Fy, and the condition says

1%2Fx+%2B+1%2Fy = 5,        (1)

%281%2Fx%29%2A%281%2Fy%29%2A3 = 18.   (2)

The last equation is equivalent to

%281%2Fx%29%2A%281%2Fy%29 = 6.      (3)

So, you need to solve the system (1),(3).

To make it easier, introduce new variables  a = 1%2Fx, b = 1%2Fy.

Then the system (1),(3) takes the form

a + b = 5,     (4)
ab    = 6.     (5)

From (4),  a = 5-b.  Substitute it into (4). You will get

(5-b)*b = 6,

5b - b^2 = 6,

b^2 - 5b + 6 = 0,

(b-2)*(b-3) = 0.


The roots are b%5B1%5D = 2   and  b%5B2%5D = 3.


1.  If  b = 2, then a = 3,  and x = 1%2F3,  y = 1%2F2.


2.  If  b = 3, then a = 2,  and x = 1%2F2,  y = 1%2F3.


Answer.  The two numbers are  1%2F2  and  1%2F3.

Solved.