Question 1171175
.


            First,  we accept the agreement that we consider only positive values of the square roots.


            After that,  we solve the problem in several steps.



<pre>
<U>Step 1</U>.  We find 6x + 5y from the first equation.


    {{{6x + 5y + sqrt(6x+5y)}}} = 72.


    For it, we introduce new variable u = {{{sqrt(6x + 5y)}}}.

    Then the equation takes the form

         u^2 + u = 72

         u^2 + u - 72 = 0

         (u+9)*(u-8) = 0.

     The roots are -9 and 8, but we accept only positive root  u= 8.


     Then we have THIS equation 

         6x + 5y = {{{8^2}}},   or

         6x + 5y = 64.    (1)


     Step 1 is complete.




<U>Step 2</U>.  We find 3x - 4y from the second equation.


    {{{3x - 4y + sqrt(3x-4y)}}} = 30.


    For it, we introduce new variable v = {{{sqrt(3x - 4y)}}}.

    Then the equation takes the form

         v^2 + v = 30

         v^2 + v - 30 = 0

         (v+6)*(v-5) = 0.

     The roots are -6 and 5, but we accept only positive root  v= 5.


     Then we have THIS equation 

         3x - 5y = {{{5^2}}},   or

         3x - 5y = 25.    (2)


     Step 2 is complete.




<U>Step 3</U>.  Solving the system (1), (2) to find x and y.


    The system is

         6x + 5y = 64.    (1)

         3x - 5y = 25.    (2)


    Solve it by ANY WAY you know / (you like) / (you prefer).


    The solution is  x = {{{127/13}}};  y = {{{14/13}}}.




<U>Step 4</U>.  Calculate  y - x.


    It is    y - x = {{{14/13}}} - {{{127/13}}} = {{{-114/13}}} = -8.


The solution is completed.


The   <U>ANSWER</U>   is   y - x = -8.
</pre>


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