SOLUTION: What is the difference (y-x) if: 6x + 5y + √(6x + 5y) = 72 3x - 4y + √(3x - 4y) = 30

Algebra ->  Square-cubic-other-roots -> SOLUTION: What is the difference (y-x) if: 6x + 5y + √(6x + 5y) = 72 3x - 4y + √(3x - 4y) = 30      Log On


   



Question 1171175: What is the difference (y-x) if:

6x + 5y + √(6x + 5y) = 72
3x - 4y + √(3x - 4y) = 30

Answer by ikleyn(52835) About Me  (Show Source):
You can put this solution on YOUR website!
.

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

            After that,  we solve the problem in several steps.


Step 1.  We find 6x + 5y from the first equation.


    6x+%2B+5y+%2B+sqrt%286x%2B5y%29 = 72.


    For it, we introduce new variable u = sqrt%286x+%2B+5y%29.

    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%5E2,   or

         6x + 5y = 64.    (1)


     Step 1 is complete.




Step 2.  We find 3x - 4y from the second equation.


    3x+-+4y+%2B+sqrt%283x-4y%29 = 30.


    For it, we introduce new variable v = sqrt%283x+-+4y%29.

    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%5E2,   or

         3x - 5y = 25.    (2)


     Step 2 is complete.




Step 3.  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%2F13;  y = 14%2F13.




Step 4.  Calculate  y - x.


    It is    y - x = 14%2F13 - 127%2F13 = -114%2F13 = -8.


The solution is completed.


The   ANSWER   is   y - x = -8.


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            SOLVED.
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