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2log5x+3log3y=8
6log5x+2log3y=2 find the values of x and y
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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).
Solved.
The method to solve such problem is to introduce new variables
and reduce the given non-linear equations to linear ones.