Question 1207662
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If k = (x + 3)/(x - 4) and k^2 - 3k = 28, find x.
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<pre>
First solve the quadratic equation for k and find its solutions for k.


    k^2 - 3k = 28

is equivalent to

    k^2 - 3k - 28 = 0.


Factor left side

    (k-7)*(k+4) = 0.


The solutions are k= 7 and k= -4.


        Now consider and solve equation  {{{(x+3)/(x-4)}}} = k   
        for two values of k: k= 7 and  k= -4.



(a)  Case k = 7.

     Now solve equation

          {{{(x+3)/(x-4)}}} = 7

     Step by step

          x+3 = 7*(x-4)

          x+3 = 7x - 28

           3 + 28 = 7x - x

            31 = 6x

             x = {{{31/6}}}.



(b)  Case k = -4.

     Now solve equation

          {{{(x+3)/(x-4)}}} = -4

     Step by step

          x+3 = (-4)*(x-4)

          x+3 = -4x + 16

           x + 4x = 16 - 3

            5x = 13

             x = {{{13/5}}}.


<U>ANSWER</U>.  Two solutions are  x = {{{31/6}}}  and  x = {{{13/5}}}.
</pre>

Solved from the beginning to the end, with complete explanations.