To add or subtract fractions, you need a common denominator. Here, we are asked to add the fractions:
[tex]\[
\frac{2}{3a-5} + \frac{3}{4a}
\][/tex]
Step 1: Find the common denominator.
The denominators are [tex]\(3a-5\)[/tex] and [tex]\(4a\)[/tex]. The least common multiple (LCM) of these denominators will be their product because they have no common factors:
[tex]\[
\text{Common denominator} = (3a - 5)(4a)
\][/tex]
Step 2: Rewrite each fraction with the common denominator.
We need to express each fraction so that they have the same denominator, [tex]\((3a-5)(4a)\)[/tex].
For the first fraction:
[tex]\[
\frac{2}{3a-5} = \frac{2 \cdot 4a}{(3a-5) \cdot 4a} = \frac{8a}{4a(3a-5)}
\][/tex]
For the second fraction:
[tex]\[
\frac{3}{4a} = \frac{3 \cdot (3a-5)}{(3a-5) \cdot 4a} = \frac{3(3a-5)}{4a(3a-5)} = \frac{9a - 15}{4a(3a-5)}
\][/tex]
Step 3: Add the rewritten fractions, combining the numerators over the common denominator:
[tex]\[
\frac{8a}{4a(3a-5)} + \frac{9a - 15}{4a(3a-5)} = \frac{8a + (9a - 15)}{4a(3a-5)}
\][/tex]
Step 4: Simplify the numerator:
[tex]\[
8a + 9a - 15 = 17a - 15
\][/tex]
Therefore, the combined fraction is:
[tex]\[
\frac{17a - 15}{4a(3a-5)}
\][/tex]
So the final answer is:
[tex]\[
\frac{17a - 15}{4a(3a-5)}
\][/tex]
