Question

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.

Answer 1

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?