Question: Consider the following equation.
$\frac{a - 6}{8} + \frac{8}{9} = \frac{a - 6}{16}$
Solve the equation.
1. **State the problem:** Solve the equation $$\frac{a - 6}{8} + \frac{8}{9} = \frac{a - 6}{16}$$.
2. **Identify the formula and rules:** This is a linear equation involving fractions. To solve it, we will eliminate the fractions by finding the least common denominator (LCD) and then isolate the variable $a$.
3. **Find the LCD:** The denominators are $8$, $9$, and $16$. The LCD of $8$, $9$, and $16$ is $144$.
4. **Multiply both sides of the equation by $144$ to clear denominators:**
$$144 \times \left( \frac{a - 6}{8} + \frac{8}{9} \right) = 144 \times \frac{a - 6}{16}$$
5. **Distribute multiplication:**
$$144 \times \frac{a - 6}{8} + 144 \times \frac{8}{9} = 144 \times \frac{a - 6}{16}$$
Simplify each term:
$$18(a - 6) + 16 \times 8 = 9(a - 6)$$
6. **Write the simplified equation:**
$$18(a - 6) + 128 = 9(a - 6)$$
7. **Expand the terms:**
$$18a - 108 + 128 = 9a - 54$$
8. **Combine like terms on the left:**
$$18a + 20 = 9a - 54$$
9. **Isolate variable terms on one side and constants on the other:**
$$18a - 9a = -54 - 20$$
$$9a = -74$$
10. **Divide both sides by 9:**
$$a = \frac{-74}{9}$$
11. **Final answer:**
$$a = -\frac{74}{9}$$