We are given the equation:

\[
\frac{x+2}{5} = \frac{10 - 2x}{3}
\]

### Step 1: Eliminate the denominators
To eliminate the denominators, multiply both sides of the equation by 15 (the least common denominator of 5 and 3).

\[
15 \times \frac{x+2}{5} = 15 \times \frac{10-2x}{3}
\]

This simplifies to:

\[
3(x+2) = 5(10 - 2x)
\]

### Step 2: Expand both sides
Now, distribute the constants on both sides.

\[
3x + 6 = 50 - 10x
\]

### Step 3: Collect like terms
Move all terms involving \( x \) to one side and constants to the other side:

\[
3x + 10x = 50 - 6
\]

\[
13x = 44
\]

### Step 4: Solve for \( x \)
Now, divide both sides by 13:

\[
x = \frac{44}{13}
\]

Thus, the solution is:

\[
x = \frac{44}{13}
\]

Would you like further details or have any questions?

Here are five questions to further explore this topic:
1. How do we determine the least common denominator in such equations?
2. Why is it important to eliminate the denominators first?
3. Can this equation have any other solutions?
4. How would the equation change if the constants were different?
5. What happens if you move all variables to the opposite side first?

**Tip:** Always check your final solution by plugging it back into the original equation to verify!

Explore how to solve the linear equation (x+2)/5 = (10-2x)/3 with a detailed step-by-step solution. This problem covers key algebraic concepts such as cross multiplication and collecting like terms, making it ideal for students in Grades 8-10.

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