Question: Solve each equation.
$$\frac{9x + 7}{2} - x + \frac{x - 2}{7} = 36$$
1. **State the problem:** Solve the equation $$\frac{9x + 7}{2} - x + \frac{x - 2}{7} = 36$$.
2. **Identify the formula and rules:** To solve this linear equation with fractions, we first find a common denominator to clear the fractions, then simplify and solve for $$x$$.
3. **Find the least common denominator (LCD):** The denominators are $$2$$ and $$7$$, so $$\text{LCD} = 14$$.
4. **Multiply every term by the LCD to clear fractions:**
$$14 \times \left(\frac{9x + 7}{2} - x + \frac{x - 2}{7}\right) = 14 \times 36$$
5. **Distribute multiplication:**
$$14 \times \frac{9x + 7}{2} - 14 \times x + 14 \times \frac{x - 2}{7} = 504$$
6. **Simplify each term:**
$$7(9x + 7) - 14x + 2(x - 2) = 504$$
7. **Expand the parentheses:**
$$7 \times 9x + 7 \times 7 - 14x + 2x - 4 = 504$$
$$63x + 49 - 14x + 2x - 4 = 504$$
8. **Combine like terms:**
$$63x - 14x + 2x + 49 - 4 = 504$$
$$51x + 45 = 504$$
9. **Isolate the variable term:**
$$51x = 504 - 45$$
$$51x = 459$$
10. **Divide both sides by 51:**
$$x = \frac{459}{51}$$
11. **Simplify the fraction:**
$$x = \frac{\cancel{459}^{9} \times 51}{\cancel{51}^{1} \times 51} = 9$$
**Final answer:** $$x = 9$$