Let's solve the equation $\frac{X}{3} - 2X + \frac{1}{3} = X - \frac{3}{5}$.

Step 1: Clear the fractions

To eliminate the fractions, we can multiply every term by the least common denominator (LCD) of 3 and 5, which is 15.

$15 \times \left(\frac{X}{3}\right) - 15 \times 2X + 15 \times \left(\frac{1}{3}\right) = 15 \times X - 15 \times \left(\frac{3}{5}\right)$

This simplifies to:

$5X - 30X + 5 = 15X - 9$

Step 2: Combine like terms

Now, combine the $X$ terms and the constants:

$5X - 30X = 15X - 9 - 5$

This simplifies to:

$-25X = 15X - 14$

Step 3: Move all terms involving $X$ to one side

Add $25X$ to both sides to get:

$0 = 40X - 14$

Step 4: Solve for $X$

Now, solve for $X$ by isolating it:

$40X = 14$

$X = \frac{14}{40}$

Simplify the fraction:

$X = \frac{7}{20}$

Final Answer:

$X = \frac{7}{20}$

Would you like more details, or do you have any questions about the solution?

Here are 5 related questions for further practice:

1. How would you solve $\frac{2X}{5} - \frac{3}{2} = \frac{X}{3} + \frac{1}{6}$?
2. Can you solve $4X - \frac{2}{3} = \frac{X}{2} + \frac{5}{4}$?
3. What is the solution to $\frac{3X}{4} - 5X + \frac{2}{3} = 2X - \frac{7}{8}$?
4. How would you approach $\frac{X}{2} + \frac{3X}{5} = \frac{X}{4} + \frac{1}{2}$?
5. Solve $\frac{X+2}{3} - X = \frac{1}{5}$.

Tip: Always clear fractions early in the equation by multiplying by the least common denominator; this simplifies the equation and reduces potential errors.