Let's solve the equation:

$-\frac{2}{5}x + 3 = \frac{2}{3}x + \frac{1}{3}$

Step 1: Eliminate the fractions

To clear the fractions, find the least common denominator (LCD) of 5 and 3, which is 15. Multiply both sides of the equation by 15:

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

Distribute 15 on both sides:

$15 \times -\frac{2}{5}x + 15 \times 3 = 15 \times \frac{2}{3}x + 15 \times \frac{1}{3}$

This simplifies to:

$-6x + 45 = 10x + 5$

Step 2: Move the variables to one side

Now, subtract $10x$ from both sides:

$-6x - 10x + 45 = 5$

Simplifying:

$-16x + 45 = 5$

Step 3: Isolate $x$

Subtract 45 from both sides:

$-16x = 5 - 45$

Simplifying:

$-16x = -40$

Step 4: Solve for $x$

Divide both sides by -16:

$x = \frac{-40}{-16}$

Simplify the fraction:

$x = \frac{40}{16} = \frac{5}{2}$

Final Answer:

$x = \frac{5}{2}$

Would you like any further clarification or details on this?

Here are some related questions to consider:

1. What happens when the equation involves more complex fractions?
2. How do you solve equations with variables on both sides?
3. Can we solve equations using decimals instead of fractions?
4. What are other methods to eliminate fractions in equations?
5. How does changing the coefficients affect the solution?

Tip: Always check for common denominators when working with fractions to simplify solving!