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Find x and y given that e^x + 3e^y=3 and e^2x - 9e^2y=6, expressing your answers as a logarithm to base e.
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Introduce two new variables u = and v = .
Then your system will be reduced to the system
u + 3v = 3, (1)
u^2 - 9v^2 = 6. (2)
Factor the second equation
(u-3v)*(u+3v) = 6
and substitute 3 instead of u+3x in the last equation. You will get the system
u + 3v = 3, (1')
u - 3v = = 2. (2')
Add the two last equations. You will get
2u = 5. Hence, u = .
Now, distract (2) from (1'). You will get
6v = 1. Hence, v = .
Thus we get
u = = -----> x = = .
v = = -----> y = = .