Let's analyze the given expressions to determine whether they are equivalent to [tex]\(-\frac{x}{y}\)[/tex]:
### Expression A: [tex]\(-\frac{-x}{y}\)[/tex]
1. The original expression [tex]\( -\frac{x}{y} \)[/tex] means we take [tex]\(\frac{x}{y}\)[/tex] and then apply a negative sign to the entire fraction.
2. For [tex]\(-\frac{-x}{y}\)[/tex], we start by considering the numerator and denominator:
- The numerator is [tex]\(-x\)[/tex], so the double negative [tex]\(-(-x)\)[/tex] simplifies to [tex]\(x\)[/tex].
3. Thus, we have [tex]\(-\frac{-x}{y} = \frac{x}{y}\)[/tex].
4. This final simplification [tex]\(\frac{x}{y}\)[/tex] is not the same as [tex]\(-\frac{x}{y}\)[/tex].
So, [tex]\(-\frac{-x}{y}\)[/tex] is not equivalent to [tex]\(-\frac{x}{y}\)[/tex].
### Expression B: [tex]\(\frac{x}{-y}\)[/tex]
1. Again, consider the original expression [tex]\( -\frac{x}{y} \)[/tex]:
- The negative sign applied to the whole fraction can be interpreted as applying to either the numerator or the denominator separately.
2. [tex]\(\frac{x}{-y}\)[/tex] places the negative sign with the denominator:
- This can be rewritten as [tex]\(-\frac{x}{y}\)[/tex].
3. Therefore, [tex]\(\frac{x}{-y}\)[/tex] simplifies directly to [tex]\(-\frac{x}{y}\)[/tex].
So, [tex]\(\frac{x}{-y}\)[/tex] is equivalent to [tex]\(-\frac{x}{y}\)[/tex].
### Conclusion
Based on our analysis, the expression [tex]\( -\frac{x}{y} \)[/tex] is equivalent to only one of the given options:
- Option B: [tex]\(\frac{x}{-y}\)[/tex]
Thus, the correct equivalent expression is only [tex]\( B \)[/tex].
