Let's solve the equation:

$\frac{2x}{3} + 1 = \frac{7x}{15} + 3$

Step 1: Eliminate the fractions by finding a common denominator

The least common denominator (LCD) of 3 and 15 is 15. Multiply every term by 15 to eliminate the denominators.

$15 \times \left(\frac{2x}{3} + 1\right) = 15 \times \left(\frac{7x}{15} + 3\right)$

This gives:

$5(2x) + 15(1) = 1(7x) + 15(3)$

Simplify the terms:

$10x + 15 = 7x + 45$

Step 2: Move all terms involving $x$ to one side

Subtract $7x$ from both sides:

$10x - 7x + 15 = 45$

$3x + 15 = 45$

Step 3: Move the constant term to the other side

Subtract 15 from both sides:

$3x = 30$

Step 4: Solve for $x$

Divide by 3:

$x = 10$

Final Answer:

$x = 10$

Would you like more details or further explanation?

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Here are 5 related questions to expand your understanding:

1. How do you solve equations with different denominators?
2. What is the least common denominator, and how is it useful?
3. How would you check if your solution is correct?
4. How do you solve equations involving multiple fractions?
5. What are the steps to isolate a variable in a linear equation?

Tip: Always check your solution by plugging it back into the original equation to ensure both sides are equal.