SOLUTION: Number 8: Number theory from Mixed Problem Solving Find 2 numbers whose sum is 56 and whose product is 783.

Number 8:  Number theory from Mixed Problem Solving



Find 2 numbers whose sum is 56 and whose product is 783.



What is asked in the problem?

 Find 2 numbers whose sum is 56 and whose product is 783.



Representation:

 Let x be the first number

     y be the second number



Equation:

 x + y = 56   (1st equation)

    xy = 783  (2nd equation)



Solve the 2 equations, solve for x in terms of y using the first equation

   x + y = 56

       x = 56 - y



Substitute x to the second equation

              xy = 783, where x = 56 - y

      56 - y (y) = 783

    56(y) - y(y) = 783

       56y - y^2 = 783

               0 = y^2 - 56y + 783

 Factor the equation y^2 - 56y + 783 = 0 to find the value of y.



            0 = y^2 - 56y + 783

            0 = (y - 29)(y - 27) 

            0 = y - 29    0 = y - 27

            y = 29        y = 27



There are 2 values for y. Substitute the values to either 

of the two equations for find values of x.



     x + y = 56, if y = 29

    x + 29 = 56

         x = 56 - 29

         x = 27



     x + y = 56, if y = 27     

    x + 27 = 56

         x = 56 - 27

         x = 29





Therefore if x = 29, y = 27. 

          if x = 27, y = 29.

There are two possibilities with the given problem.