Let's simplify the expression:

\[
\frac{4x - y}{3} - \frac{x - 3y}{2}
\]

### Step 1: Find a common denominator
The denominators are 3 and 2. The least common denominator (LCD) is 6.

### Step 2: Rewrite both fractions with the common denominator
To make both fractions have the denominator of 6, we multiply the first fraction by 2 (top and bottom) and the second fraction by 3 (top and bottom):

\[
\frac{4x - y}{3} = \frac{2(4x - y)}{6} = \frac{8x - 2y}{6}
\]
\[
\frac{x - 3y}{2} = \frac{3(x - 3y)}{6} = \frac{3x - 9y}{6}
\]

### Step 3: Combine the fractions
Now we subtract the two fractions:

\[
\frac{8x - 2y}{6} - \frac{3x - 9y}{6} = \frac{(8x - 2y) - (3x - 9y)}{6}
\]

Simplify the numerator:

\[
(8x - 2y) - (3x - 9y) = 8x - 2y - 3x + 9y = (8x - 3x) + (-2y + 9y) = 5x + 7y
\]

So the expression becomes:

\[
\frac{5x + 7y}{6}
\]

This is the simplified form.

Would you like more details on this? 

Here are 5 related questions:
1. How do you solve similar equations with different denominators?
2. Can you explain how to find the least common denominator (LCD)?
3. What are some techniques to simplify algebraic expressions?
4. How can you solve an equation when you have three terms?
5. What are some applications of algebraic fractions in real life?

**Tip:** Always look for a common denominator when subtracting or adding fractions—it simplifies the process!

Learn how to simplify the algebraic expression 4x - y / 3 - x - 3y / 2 with step-by-step guidance. This problem explores concepts like finding the least common denominator, rewriting fractions, and combining terms, making it suitable for students in Grades 8-10.

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