We have the equation:

\[
\frac{2x}{5} + \frac{1}{2} = \frac{x}{3} + \frac{5}{8}
\]

To solve this, let's follow the steps:

### Step 1: Eliminate the fractions by finding the least common denominator (LCD).
The denominators are \( 5, 2, 3, \) and \( 8 \). The LCD of these numbers is 120.

### Step 2: Multiply both sides of the equation by 120 to eliminate the fractions:

\[
120 \left( \frac{2x}{5} + \frac{1}{2} \right) = 120 \left( \frac{x}{3} + \frac{5}{8} \right)
\]

Simplifying each term:

\[
120 \times \frac{2x}{5} = 48x, \quad 120 \times \frac{1}{2} = 60, \quad 120 \times \frac{x}{3} = 40x, \quad 120 \times \frac{5}{8} = 75
\]

So, the equation becomes:

\[
48x + 60 = 40x + 75
\]

### Step 3: Simplify and solve for \(x\).

Subtract \(40x\) from both sides:

\[
48x - 40x + 60 = 75
\]

This simplifies to:

\[
8x + 60 = 75
\]

### Step 4: Isolate \(x\).

Subtract 60 from both sides:

\[
8x = 15
\]

Now, divide by 8:

\[
x = \frac{15}{8}
\]

Thus, the solution is:

\[
x = \frac{15}{8}
\]

Would you like further details or explanations?

Here are 5 related questions to expand on this:

1. How can we check the solution for \(x = \frac{15}{8}\)?
2. What is the process of finding the least common denominator in a set of fractions?
3. Can this equation be solved using substitution methods?
4. How do we handle fractions with variables in both sides of an equation?
5. What if we had decimals instead of fractions in this equation?

**Tip:** Always simplify equations by clearing fractions or decimals first to make calculations easier.

This page explains how to solve the linear equation ²x/5 + ½ = x/³ + ⅝ step by step. Learn how to find the least common denominator and simplify the equation to isolate the variable. This problem is suitable for students in Grades 8-10, covering key algebraic concepts and solving techniques.

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