Question: Solve for $x$.
$$\frac{x + 8}{5} - \frac{x - 5}{3} = 5$$
1. **State the problem:** Solve the linear equation with fractions:
$$\frac{x + 8}{5} - \frac{x - 5}{3} = 5$$
2. **Identify the formula and rules:** To solve equations with fractions, find a common denominator to eliminate fractions by multiplying both sides.
3. **Find the least common denominator (LCD):** The denominators are $5$ and $3$, so the LCD is $15$.
4. **Multiply both sides by the LCD to clear fractions:**
$$15 \times \left(\frac{x + 8}{5} - \frac{x - 5}{3}\right) = 15 \times 5$$
5. **Distribute multiplication:**
$$15 \times \frac{x + 8}{5} - 15 \times \frac{x - 5}{3} = 75$$
6. **Simplify each term:**
$$\cancel{15} \times \frac{x + 8}{\cancel{5}} = 3(x + 8)$$
$$\cancel{15} \times \frac{x - 5}{\cancel{3}} = 5(x - 5)$$
So the equation becomes:
$$3(x + 8) - 5(x - 5) = 75$$
7. **Expand the parentheses:**
$$3x + 24 - 5x + 25 = 75$$
8. **Combine like terms:**
$$3x - 5x + 24 + 25 = 75$$
$$-2x + 49 = 75$$
9. **Isolate the variable term:**
$$-2x = 75 - 49$$
$$-2x = 26$$
10. **Divide both sides by $-2$ to solve for $x$:**
$$x = \frac{26}{-2}$$
$$x = -13$$
**Final answer:**
$$\boxed{-13}$$