To solve the equation:

\[
\frac{2x + 1}{8} + \frac{x - 6}{12} = 10
\]

### Step 1: Find a common denominator
The least common denominator (LCD) of 8 and 12 is 24. So, rewrite both fractions with 24 as the denominator:

\[
\frac{2x + 1}{8} = \frac{3(2x + 1)}{24} = \frac{6x + 3}{24}
\]
\[
\frac{x - 6}{12} = \frac{2(x - 6)}{24} = \frac{2x - 12}{24}
\]

Now the equation becomes:

\[
\frac{6x + 3}{24} + \frac{2x - 12}{24} = 10
\]

### Step 2: Combine the fractions
Now that the denominators are the same, combine the numerators:

\[
\frac{(6x + 3) + (2x - 12)}{24} = 10
\]

Simplify the numerator:

\[
\frac{6x + 2x + 3 - 12}{24} = 10
\]
\[
\frac{8x - 9}{24} = 10
\]

### Step 3: Eliminate the denominator
Multiply both sides by 24 to eliminate the denominator:

\[
8x - 9 = 240
\]

### Step 4: Solve for \( x \)
Add 9 to both sides:

\[
8x = 249
\]

Divide both sides by 8:

\[
x = \frac{249}{8} = 31.125
\]

### Final Answer:
\[
x = 31.125
\]

Would you like any further details or have any questions?

Here are five related questions:
1. How would the solution change if the constant on the right-hand side was different?
2. Can you solve a similar equation but with more complex fractions?
3. How would you approach the problem if the fractions had variables in their denominators?
4. What is the geometric interpretation of this equation?
5. How do you solve this equation using decimals instead of fractions?

**Tip:** Always check for common denominators when adding or subtracting fractions in equations.

Learn how to solve the linear equation (2X +1)/8 + (x-6)/12 = 10 using the least common denominator method. This solution includes step-by-step instructions to combine fractions, eliminate denominators, and solve for x. Ideal for students in Grades 8-10.

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