SOLUTION: 2log5x+3log3y=8 6log5x+2log3y=2 find the values of x and y

Introduce new variables  u = log(5x),  v = log(3y).

Then the original systems of equations takes the form

    2u + 3v = 8    (1)

    6u + 2v = 2    (2)


Solve using elimination

    6u + 9v = 24
    6u + 2v =  2
---------------------------Subtract

         7v = 22  --->  v = .


Then from equation (1) we have

    2u + 66/7 = 8

    2u = 8 - 66/7 = -10/7 --->  u = .


Now, log(5x) = u =   --->  5x = 10^(-5/7)  --->  x = (1/5)*10^(-5/7).

     log(3y) = v = 22/7  --->  3y = 10^(22/7)  --->  y = (1/3)*10^(22/7).


ANSWER.  x = (1/5)*10^(-5/7),  y = (1/3)*10^(22/7).