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The question asks to simplify the following expression:

\[
\frac{2x + 1}{3} - \frac{4x - 1}{7} + 1
\]

Let's go through the steps to simplify it:

1. **Find a Common Denominator**: The denominators are 3 and 7, so the least common multiple (LCM) is 21.

2. **Rewrite Each Term with a Denominator of 21**:
   - For the first fraction: 
     \[
     \frac{2x + 1}{3} = \frac{7(2x + 1)}{21} = \frac{14x + 7}{21}
     \]
   - For the second fraction:
     \[
     \frac{4x - 1}{7} = \frac{3(4x - 1)}{21} = \frac{12x - 3}{21}
     \]
   - The constant \(1\) can be written as:
     \[
     1 = \frac{21}{21}
     \]

3. **Combine the Fractions**:
   \[
   \frac{14x + 7}{21} - \frac{12x - 3}{21} + \frac{21}{21} = \frac{(14x + 7) - (12x - 3) + 21}{21}
   \]

4. **Simplify the Numerator**:
   - Distribute and combine like terms:
     \[
     = \frac{14x + 7 - 12x + 3 + 21}{21} = \frac{(14x - 12x) + (7 + 3 + 21)}{21} = \frac{2x + 31}{21}
     \]

So, the simplified form of the expression is:

\[
\frac{2x + 31}{21}
\]

Would you like further explanation on any of these steps, or do you have any additional questions?

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### Related Questions
1. How do you find the least common multiple (LCM) of two numbers?
2. How can we simplify expressions involving fractions with different denominators?
3. What other methods can be used to simplify algebraic fractions?
4. How would this problem change if there were more terms with different denominators?
5. How can we check if our simplified result is correct?

### Tip
When dealing with algebraic fractions, always start by finding a common denominator to make combining terms easier.

Simplify the expression: (2x + 1) / 3 - (4x - 1) / 7 + 1

Simplify the algebraic expression (2x + 1) / 3 - (4x - 1) / 7 + 1 by finding a common denominator and combining terms. Detailed solution for Grades 9-10 students to enhance algebra and fraction simplification skills.

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