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Wednesday, 2 December 2015

Solve the pair of simultaneous equations \log(x + y) = 0 ...(1) \quad\quad2 \log x = \log (y - 1) ...(2)

Solve the pair of simultaneous equations 

    \log(x + y) = 0             ...(1) 

    2 \log x = \log (y - 1)    ...(2) 

SOLUTION

\log (x+y)=0 in index form becomes x+y = 10^0 and so 

                     x+y=1      (1)   

In (2) we see that the domain of x and y is restricted by x>0 and y-1>0 (since the log function has domain positive numbers only)

           so we must have x>0 and y>1

Now using log laws on (2) and converting to index form we have 

                 \log (x^2) = \log(y-1) 

                \log(x^2) - \log(y-1) = 0 

               \log( x^2/(y-1) )  = 0

              x^2/(y-1) = 10^0             (index form of previous line) 

             x^2/(y-1) = 1

            x^2 = y - 1

            y = x^2 + 1    (2) 

So we now have the two simultaneous equations to solve 

    x+y=1             (1)   
    y = x^2 + 1     (2) 
(but keeping in mind that x>0, y>1 must hold also) 

so,              x + x^2 +1 = 1 
                 
                  x^2 + x = 0 

                 x ( x + 1 )= 0

               so this gives x = 0, -1 
             but we must have x>0 and both of these solutions fail that condition. 

             So, we conclude that there are NO SOLUTIONS! 

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