To simplify the given expression [tex]\(\frac{3}{x} + \frac{2x}{y}\)[/tex], follow these steps:
1. Identify the given fractions: The expression consists of two fractions: [tex]\(\frac{3}{x}\)[/tex] and [tex]\(\frac{2x}{y}\)[/tex].
2. Find a common denominator: To combine the fractions, we need to express them with a common denominator. The denominators are [tex]\(x\)[/tex] and [tex]\(y\)[/tex]. The common denominator for [tex]\(x\)[/tex] and [tex]\(y\)[/tex] would be [tex]\(xy\)[/tex].
3. Rewrite each fraction with the common denominator:
- For [tex]\(\frac{3}{x}\)[/tex], multiply the numerator and the denominator by [tex]\(y\)[/tex]:
[tex]\[
\frac{3}{x} = \frac{3 \cdot y}{x \cdot y} = \frac{3y}{xy}
\][/tex]
- For [tex]\(\frac{2x}{y}\)[/tex], multiply the numerator and the denominator by [tex]\(x\)[/tex]:
[tex]\[
\frac{2x}{y} = \frac{2x \cdot x}{y \cdot x} = \frac{2x^2}{xy}
\][/tex]
4. Combine the fractions: Now that both fractions have the same denominator, combine them:
[tex]\[
\frac{3y}{xy} + \frac{2x^2}{xy} = \frac{3y + 2x^2}{xy}
\][/tex]
5. Express the combined fraction in its simplest form: Look to see if the numerator or denominator can be factored further or reduced, but in this case, the combined fraction is already in its simplest form.
Thus, the simplified expression is:
[tex]\[
\boxed{\frac{2x}{y} + \frac{3}{x}}
\][/tex]
