To express the sum [tex]\(\frac{1}{x} + \frac{1}{y}\)[/tex] in its simplest form, we follow these steps:
1. Find a common denominator:
- The denominators of the fractions are [tex]\(x\)[/tex] and [tex]\(y\)[/tex].
- The least common multiple (LCM) of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] is [tex]\(xy\)[/tex].
2. Rewrite each fraction with the common denominator:
- For [tex]\(\frac{1}{x}\)[/tex]:
[tex]\[
\frac{1}{x} = \frac{y}{xy}
\][/tex]
- For [tex]\(\frac{1}{y}\)[/tex]:
[tex]\[
\frac{1}{y} = \frac{x}{xy}
\][/tex]
3. Add the fractions:
- Now that both fractions have the same denominator, we can add them:
[tex]\[
\frac{y}{xy} + \frac{x}{xy} = \frac{y + x}{xy}
\][/tex]
4. Simplify the expression:
- The numerator is [tex]\(y + x\)[/tex] and the denominator is [tex]\(xy\)[/tex].
- This fraction is already in its simplest form since there are no common factors to further simplify.
So, the sum [tex]\(\frac{1}{x} + \frac{1}{y}\)[/tex] in its simplest form is:
[tex]\[
\frac{y + x}{xy}
\][/tex]
Thus, the correct choice is:
A. [tex]\(\frac{y+x}{xy}\)[/tex]
