Question 98759
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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.