Question

Solve for yy. Express your answer as a proper or improper fraction in simplest terms.
start fraction, 5, divided by, 6, end fraction, minus, start fraction, 3, divided by, 8, end fraction, y, equals, start fraction, 2, divided by, 3, end fraction
6
5
​
−
8
3
​
y=
3
2
​

Answer 1

To solve the equation

\[
\frac{5}{6} - \frac{3}{8}y = \frac{2}{3}
\]

we first isolate the term involving \(y\). Start by moving \(\frac{5}{6}\) to the right side of the equation:

\[
-\frac{3}{8}y = \frac{2}{3} - \frac{5}{6}
\]

Next, we need to find a common denominator to subtract the two fractions on the right. The least common multiple (LCM) of 3 and 6 is 6. We rewrite \(\frac{2}{3}\) with a denominator of 6:

\[
\frac{2}{3} = \frac{2 \times 2}{3 \times 2} = \frac{4}{6}
\]

Now, we can perform the subtraction:

\[
-\frac{3}{8}y = \frac{4}{6} - \frac{5}{6} = \frac{4 - 5}{6} = \frac{-1}{6}
\]

Now we will eliminate the negative sign by multiplying both sides by \(-1\):

\[
\frac{3}{8}y = \frac{1}{6}
\]

To solve for \(y\), multiply both sides by the reciprocal of \(\frac{3}{8}\):

\[
y = \frac{1}{6} \times \frac{8}{3}
\]

This simplifies as follows:

\[
y = \frac{1 \times 8}{6 \times 3} = \frac{8}{18}
\]

Now, simplifying \(\frac{8}{18}\):

The greatest common divisor (GCD) of 8 and 18 is 2, so we divide both the numerator and the denominator by 2:

\[
y = \frac{8 \div 2}{18 \div 2} = \frac{4}{9}
\]

Therefore, the solution for \(y\) is:

\[
\boxed{\frac{4}{9}}
\]