To determine which expression is equivalent to the given complex fraction, let's simplify the fraction step by step.
The given complex fraction is:
[tex]\[
\frac{2 - \frac{1}{y}}{3 + \frac{1}{y}}
\][/tex]
To eliminate the fractions within the numerator and the denominator, let's find a common denominator for the terms in both the numerator and the denominator.
1. Combine terms in the numerator and denominator:
\b
Numerator: [tex]\(2 - \frac{1}{y} = \frac{2y - 1}{y}\)[/tex]
Denominator: [tex]\(3 + \frac{1}{y} = \frac{3y + 1}{y}\)[/tex]
Now the complex fraction becomes:
[tex]\[
\frac{\frac{2y - 1}{y}}{\frac{3y + 1}{y}}
\][/tex]
2. Simplify the complex fraction by multiplying both the numerator and the denominator of the large fraction by [tex]\( y \)[/tex]:
[tex]\[
\frac{\frac{2y - 1}{y} \cdot y}{\frac{3y + 1}{y} \cdot y} = \frac{2y - 1}{3y + 1}
\][/tex]
Thus, the simplified form of the given complex fraction is:
[tex]\[
\frac{2y - 1}{3y + 1}
\][/tex]
Now, let's compare this result with the given choices:
[tex]\[
\frac{3 y+1}{2 y-1}
\][/tex]
[tex]\[
\frac{(2 y-1)(3 y+1)}{y^2}
\][/tex]
[tex]\[
\frac{y^2}{(2 y-1)(3 y+1)}
\][/tex]
[tex]\[
\frac{2 y-1}{3 y+1}
\][/tex]
The simplified expression [tex]\(\frac{2y - 1}{3y + 1}\)[/tex] exactly matches the fourth choice. Therefore, the correct answer is:
[tex]\(\frac{2 y-1}{3 y+1}\)[/tex]
So, the expression equivalent to the given complex fraction is:
[tex]\[
\boxed{4}
\][/tex]
