To subtract the given fractions, we first need to combine them over a common denominator.
Given fractions are:
[tex]\[ \frac{5c - 6d}{8c} - \frac{2c - 8d}{8c} \][/tex]
Since both fractions have the same denominator, [tex]\(8c\)[/tex], we can directly subtract the numerators:
[tex]\[ \frac{(5c - 6d) - (2c - 8d)}{8c} \][/tex]
First, distribute the negative sign through the second numerator:
[tex]\[ \frac{5c - 6d - 2c + 8d}{8c} \][/tex]
Next, combine the like terms in the numerator:
[tex]\[ \frac{(5c - 2c) + (-6d + 8d)}{8c} \][/tex]
[tex]\[ \frac{3c + 2d}{8c} \][/tex]
Now, we need to simplify the expression if possible. Since there is no common factor between the terms in the numerator and the denominator (other than the [tex]\(c\)[/tex] term which appears in a product), we separate the fraction:
[tex]\[ \frac{3c}{8c} + \frac{2d}{8c} \][/tex]
Now, simplify each term separately:
[tex]\[ \frac{3c}{8c} = \frac{3}{8} \][/tex]
[tex]\[ \frac{2d}{8c} = \frac{2d}{8c} = \frac{d}{4c} \][/tex]
Combining these results, we get the simplified expression:
[tex]\[ \frac{3}{8} + \frac{d}{4c} \][/tex]
This is the simplest form of the given expression.
