We are given the equation:

$\frac{X}{3} + 1 = \frac{7X}{15} + 3$

Step 1: Eliminate the fractions

To eliminate the fractions, we multiply through by the least common denominator (LCD) of 3 and 15, which is 15:

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

This gives:

$5X + 15 = 7X + 45$

Step 2: Simplify the equation

Now, let's move all the terms involving $X$ to one side and the constants to the other side:

$5X - 7X = 45 - 15$

$-2X = 30$

Step 3: Solve for $X$

Now, divide both sides by $-2$ to solve for $X$:

$X = \frac{30}{-2} = -15$

Final Answer:

The solution to the equation is $X = -15$.

Would you like a detailed breakdown of any step or have questions about this solution?

Here are some related questions for practice:

1. Solve: $\frac{2X}{4} + 3 = \frac{5X}{6} + 7$.
2. Solve: $\frac{X}{2} - 4 = \frac{3X}{8} + 2$.
3. Solve for $X$: $\frac{X}{5} + 2 = \frac{4X}{10} + 1$.
4. What is the solution to: $\frac{3X}{7} - 2 = \frac{X}{14} + 5$?
5. Solve for $X$: $\frac{X}{4} + 6 = \frac{5X}{12} - 3$.

Tip: When solving equations with fractions, multiplying through by the least common denominator simplifies the equation and helps eliminate the fractions.