What two numbers whose product is 4 and the sum of whose reciprocals is 65/56.

Question

The math word problems I need help with thank you all for being there

Answer 1

xy = 4 => y = 4/x => 1/y = x/4

 

1/x + 1/y = 65/56

 

1/x + x/4 = 65/56

 

Multiply both sides by 56x:

 

56 + 14x^2 = 65x

 

14x^2 - 65x + 56 = 0

 

[I chose to find Vertex Form and solve that.]

 

h = -b/2a = - -65/(2*14) = 65/28

 

k = c - ah^2 = 56 - 14(65/28)^2

= (56^2 - 65^2)/56 = (56-65)(56+65)/56 = -9(121)/56

 

x = h ± √(-k/a) = 65/28 ± √(- -9(121)/(56*14))

 

= 65/28 ± √(3^2(11^2)/(4*14^2))

 

= 65/28 ± 3*11/(2*14) = (65 ± 33)/28

 

x = 7/2 or 8/7

 

y = 4/x = 8/7 or 7/2

 

So the two numbers are 8/7 and 7/2.

 

check:

 

xy =? 4

(8/7)(7/2) =? 4

4 = 4   √

 

1/x + 1/y =? 65/56

1/(8/7) + 1/(7/2) =? 65/56

7/8 + 2/7 =? 65/56

7*7/(8*7) + 2*8/(7*8) =? 65/56

(49 + 16)/56 =? 65/56

65/56 = 65/56  √

Answer 2

 XY = 4

 

 1/ X + 1/ Y = 65/ 56

 

 

   X = 4 / Y 

 

    Substitute in 2nd equation:

 

       Y/ 4 +  1 / Y = 65 / 56

 

 

      Y^2 + 4 = 65/ 56

        4 Y 

      

       56 Y ^2 + 224 = 260 Y

 

        14 Y ^2 - 65 Y - 56 = 0

    

       14Y^2 - 16 Y - 49 Y - 56 =0

 

        2 Y ( 7 Y - 8 ) - 7 (  7Y - 8 ) =0

 

         ( 7Y - 8 ) ( 2Y - 7 ) = 0

 

        7y - 8 = 0     Y = 8/7      X = 4 / ( 8/7) = 7/2

 

          Test:

 

             7/8 + 2 /7 =  ( 49 + 16) /56 = 65/ 56

 

              True:

 

        2Y - 7 = 0      Y = 7/2      X = 8/7

 

                    This will work too, only X, y change their position.

 

           

 

 

Answer 3

Hi Willie,

 

Let twonumbers are x & y.

x*y=4...................Eq-1

1/x+1/y=65/56.....Eq-2

Now from Eq-1, y=4/x

substituting value of y in Eq-2,

we get,1/x+1/4/x=65/56

           1/x+x/4=65/56

            4+x^2=65/56*4x

            x^2+4=65/14 *x

            x^2-65x/14+4=0

           14x^2-65x+56=0

           14x^2-49x-16x+56=0

           7x(2x-7)-8(2x-7)=0

            (2x-7)(7x-8)=0

            2x-7=0  or 7x-8=0

             x=7/2 or x=8/7

             y=4/x=4/7/2=8/7  or  y=4/8/7=7/2.

Ans: two numbers are 7/2 & 8/7.

 

Meena from Strongsville,OH