Let's solve the given equation:

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

Step 1: Eliminate the fractions

We can clear the fractions by multiplying the entire equation by the least common denominator (LCD) of all the fractions. The LCD of 5 and 3 is 15. So, multiply each term by 15:

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

Simplifying:

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

Step 2: Move all the terms involving $x$ to one side and constant terms to the other side

Move the terms involving $x$ to the left-hand side and constants to the right-hand side:

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

Simplifying:

$-16x = -40$

Step 3: Solve for $x$

Now divide both sides by $-16$:

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

Thus, the solution is:

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

Would you like more details on any step or further clarification?

Relative Questions:

1. How do you find the least common denominator (LCD) for fractions?
2. Why do we move terms involving $x$ to one side and constants to the other side?
3. How can you check your solution once you find $x$?
4. What happens if both sides of the equation contain fractions with different denominators?
5. Can you solve this equation using a different method?

Tip:

When working with equations that have fractions, multiplying by the least common denominator helps eliminate fractions and simplify the problem.