Question: Solve for $x$ and simplify the answer fully.
$$\frac{x - 8}{x - 2} = \frac{7}{8}$$
1. **State the problem:** Solve the equation $$\frac{x - 8}{x - 2} = \frac{7}{8}$$ for $x$ and simplify fully.
2. **Use the cross-multiplication formula:** When two fractions are equal, their cross products are equal:
$$ (x - 8) \times 8 = (x - 2) \times 7 $$
3. **Multiply both sides:**
$$ 8(x - 8) = 7(x - 2) $$
4. **Distribute the constants:**
$$ 8x - 64 = 7x - 14 $$
5. **Isolate $x$ terms on one side and constants on the other:**
$$ 8x - 7x = -14 + 64 $$
6. **Simplify both sides:**
$$ x = 50 $$
7. **Check for restrictions:** The denominator $x - 2$ cannot be zero, so $x \neq 2$. Since $x = 50$ does not violate this, it is valid.
**Final answer:**
$$ x = 50 $$