To simplify the given expression:

\[
\frac{x + 3y}{4} + \frac{3x - y}{6}
\]

### Step-by-step simplification:

1. **Find the common denominator** between 4 and 6, which is 12.
2. **Rewrite each fraction with a denominator of 12**:

   \[
   \frac{x + 3y}{4} = \frac{3(x + 3y)}{12} = \frac{3x + 9y}{12}
   \]
   \[
   \frac{3x - y}{6} = \frac{2(3x - y)}{12} = \frac{6x - 2y}{12}
   \]

3. **Combine the two fractions** over a common denominator:

   \[
   \frac{3x + 9y}{12} + \frac{6x - 2y}{12} = \frac{(3x + 9y) + (6x - 2y)}{12}
   \]

4. **Simplify the numerator**:

   \[
   (3x + 9y) + (6x - 2y) = 3x + 6x + 9y - 2y = 9x + 7y
   \]

5. **Final simplified expression**:

   \[
   \frac{9x + 7y}{12}
   \]

Thus, the simplified form of the expression is:

\[
\frac{9x + 7y}{12}
\]

Would you like more details or have any questions?

### Related Questions:
1. How do you find a common denominator for more complex fractions?
2. How would this change if there were different variables in the numerators?
3. Can you further simplify \(\frac{9x + 7y}{12}\) under specific conditions?
4. What happens if either \(x\) or \(y\) is zero in this expression?
5. How do you add algebraic fractions with multiple variables?

### Tip:
When simplifying algebraic fractions, always check for a common denominator before adding or subtracting them.

Learn how to simplify the algebraic expression (x + 3y) / 4 + (3x - y) / 6 using the common denominator method. This step-by-step solution involves adding algebraic fractions and simplifying the result. Ideal for students in Grades 8-10.

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