how to isolate y from 3x-6y+ln|3x-6y|=5(x+C)

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You should come up with the following substitution such that:

`3x-6y = t => 3x = t+6y => x = (t+6y)/3 => 5x = 5(t+6y)/3`

You need to substitute t for `3x-6y`  such that:

`t + ln |t| = 5(t+6y)/3 + c`

`t - 5t/3 + ln t = 2y => -2t/3 + ln t = 2y => -2t/3*ln e + ln t = 2y`

`ln e^(-2t/3) + ln t = 2y`

Using the logarithmic identities yields:

`ln t*e^(-2t/3) = 2y => 2y = (ln t*e^(-2t/3))`

You need to substitute `3x - 6y`  for t such that:

`2y = ln (3x - 6y)*e^-2(3x - 6y)/3 => y = (ln (3x - 6y)*e^-2(3x - 6y)/3)/2`

Hence, evaluating y yields that y cannot be isolated under the given conditions.

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