Sure, let's take a detailed, step-by-step approach to simplify the given expression:
Given:
[tex]\[
\frac{5x - y}{10xy} - \frac{x - 5y}{10xy}
\][/tex]
Step 1: Combine the fractions since they have the same denominator.
[tex]\[
\frac{(5x - y) - (x - 5y)}{10xy}
\][/tex]
Step 2: Distribute the negative sign in the numerator.
[tex]\[
(5x - y) - (x - 5y) = 5x - y - x + 5y
\][/tex]
Step 3: Combine like terms in the numerator.
[tex]\[
5x - x + 5y - y = 4x + 4y
\][/tex]
Step 4: Simplify the expression.
[tex]\[
\frac{4x + 4y}{10xy}
\][/tex]
Step 5: Factor out the common factor of 4 from the numerator.
[tex]\[
\frac{4(x + y)}{10xy}
\][/tex]
Step 6: Simplify the fraction by dividing both the numerator and the denominator by 2.
[tex]\[
\frac{2(x + y)}{5xy}
\][/tex]
Step 7: Break down into two separate fractions.
[tex]\[
\frac{2(x + y)}{5xy} = \frac{2x}{5xy} + \frac{2y}{5xy}
\][/tex]
Step 8: Further simplify each separate fraction.
[tex]\[
\frac{2x}{5xy} = \frac{2}{5y}
\][/tex]
[tex]\[
\frac{2y}{5xy} = \frac{2}{5x}
\][/tex]
Thus, combining these results:
[tex]\[
\frac{2}{5y} + \frac{2}{5x}
\][/tex]
Therefore, the simplified expression is:
[tex]\[
\frac{2}{5y} + \frac{2}{5x}
\][/tex]
So, the correct answer to part 5 is (B) [tex]\(\frac{2x + 2y}{5xy}\)[/tex].
