Certainly! Let's break down the process to add the fractions, simplify them, and understand the given answer choices.
Given the problem, we are to add the following two fractions:
[tex]\[ -\frac{4}{y} + \left(-\frac{7}{8y}\right) \][/tex]
### Step 1: Find a common denominator
The denominators are [tex]\( y \)[/tex] and [tex]\( 8y \)[/tex]. To add the fractions, we need a common denominator. The least common denominator (LCD) is [tex]\( 8y \)[/tex].
### Step 2: Rewrite each fraction with the common denominator
We can rewrite [tex]\(-\frac{4}{y}\)[/tex] with the common denominator [tex]\( 8y \)[/tex]:
[tex]\[ -\frac{4}{y} = -\frac{4 \cdot 8}{y \cdot 8} = -\frac{32}{8y} \][/tex]
### Step 3: Combine the fractions
Now we add the two fractions:
[tex]\[ -\frac{32}{8y} + \left(-\frac{7}{8y}\right) = \frac{-32 - 7}{8y} = \frac{-32 - 7y}{8y} \][/tex]
### Step 4: Simplify the fraction
The combined fraction is:
[tex]\[ \frac{-32 - 7y}{8y} \][/tex]
This fraction is already simplified. Let's check if it matches any answer choices. The given options include:
1. [tex]\(-\frac{15}{2y}\)[/tex]
2. [tex]\(\frac{-32 - 7y}{8y}\)[/tex]
3. [tex]\(-\frac{39}{8y}\)[/tex]
4. [tex]\(-3 \frac{1}{8}\)[/tex]
Our simplified result [tex]\(\frac{-32 - 7y}{8y}\)[/tex] matches exactly with the third answer option [tex]\(\frac{-32 - 7y}{8y}\)[/tex].
### Step 5: Convert to a mixed number, if needed
Let's check if the result can be expressed as a mixed number in a simplified form. Observe that the expression [tex]\(\frac{-32 - 7y}{8y}\)[/tex] is not typically converted to a mixed number since it involves both a constant term and a linear term in 'y'.
### Conclusion
The correct answer is:
[tex]\[ \frac{-32 - 7y}{8y} \][/tex]
