Let's solve the equation \( \frac{1}{3}x + \frac{2}{5} = \frac{1}{5}x - \frac{2}{5} \).

### Step 1: Eliminate the fractions
To eliminate the fractions, find the least common denominator (LCD) for the fractions. The LCD of 3 and 5 is 15. Multiply every term in the equation by 15:

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

### Step 2: Simplify the equation
Simplify each term after multiplication:

\[
5x + 6 = 3x - 6
\]

### Step 3: Isolate the variable
To isolate \(x\), move all the terms involving \(x\) to one side of the equation and the constant terms to the other side:

\[
5x - 3x = -6 - 6
\]

\[
2x = -12
\]

### Step 4: Solve for \(x\)
Finally, divide both sides by 2 to find the value of \(x\):

\[
x = -6
\]

So, the solution is \( x = -6 \).

Would you like more details on any step, or do you have any questions?

### 5 Related Questions:
1. How do you solve equations with multiple variables?
2. What are the methods to solve quadratic equations?
3. How can you solve equations with fractions using cross multiplication?
4. What are some common mistakes to avoid when solving algebraic equations?
5. How do you graph linear equations once you've solved them?

### Tip:
Always check your solution by substituting it back into the original equation to ensure it satisfies the equation.

Learn how to solve the equation 1/3x + 2/5 = 1/5x - 2/5 with a detailed step-by-step solution. This problem involves equations with fractions and isolation of variables. Suitable for students in Grades 7-9.

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