The Cyclic Method

Consider the following equations:

                Ax + By = C

                Dx + Ey = F

To determine x :

                        Numerator = BF - CE

                        Denominator = BD - AE 

 x = ( BF - CE)/ ( BD-AE )

To determine y :

                        Numerator = CD - AF

                        Denominator = BD - AE

 y = ( CD - AF)/ ( BD - AE )                

Let us consider some examples :

1)    x -  y =  7

      5x - 2y = 42

 Using the above formulae we get,

      x =  ( -1* 42 - 7 * 2) / ( -1* 5 - 2 *1 )  =  (-56)/ (-7)  = 8

      y =  ( 7 * 5 - 42 * 1 ) / ( -1 * 5 - 2 * 1)  =  (-7) / (-7)  = 1

 Therefore  x = 8  and  y = 1

2)    5x - 3y = 11

        6x - 5y = 9

Using the above formulae we get,

    x = ( -3 * 9 - (-5)* 11 ) / ( -3 * 6 - 5 * (-5)) = 28/7 = 4

    y = ( 11 * 6 - 5 * 9 ) / ( -3 * 6 - 5 * (-5)) =  21/7 = 3

Therefore x = 4 and  y = 3

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