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Mathematics
OpenStudy (anonymous):

Here's an example: 1/2(y-1/6)+2/3=5/6+1/3(1/2-3y) I get y=5/6 and that's wrong.

OpenStudy (shadowfiend):

So we have: \[\frac{1}{2}\left(y - \frac{1}{6}\right) + \frac{2}{3} = \frac{5}{6} + \frac{1}{3}\left(\frac{1}{2} - 3y\right) \] First off, let's distribute the fractions outside the parentheses: \[\frac{1}{2}y - \frac{1}{12} + \frac{2}{3} = \frac{5}{6} + \frac{1}{6} - \frac{3}{3}y\] Then we can combine like terms: \[\begin{align} \frac{1}{2}y - \frac{1}{12} + \frac{8}{12} &= \frac{6}{6} - y\\ \frac{1}{2}y - \frac{7}{12} &= 1 - y \end{align}\] Then we can move all the \(y\)s to the left and all the non-\(y\) terms to the right: \[\frac{1}{2}y + y = 1 + \frac{7}{12} \] If we combine like terms again: \[\begin{align} \frac{1}{2}y + \frac{2}{2}y &= \frac{12}{12} + \frac{7}{12}\\ \frac{3}{2}y &= \frac{19}{12} \end{align}\] And then we can solve for y: \[\begin{align} 3y &= 2\frac{19}{12}\\ &= \frac{19}{6}\\ y &= \frac{\frac{19}{6}}{3}\\ &= \frac{19}{6\cdot 3}\\ &= \frac{19}{18} \end{align}\] Make sense?

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