1. **Solve the equation** Given: $$\frac{x - 5}{x - 2} = \frac{x - 6}{x - 4}$$ 2. **Cross multiply to eliminate denominators:** $$ (x - 5)(x - 4) = (x - 6)(x - 2) $$ 3. **Expand both sides:** $$ x^2 - 4x - 5x + 20 = x^2 - 2x - 6x + 12 $$ $$ x^2 - 9x + 20 = x^2 - 8x + 12 $$ 4. **Subtract $x^2$ from both sides:** $$ \cancel{x^2} - 9x + 20 = \cancel{x^2} - 8x + 12 $$ $$ -9x + 20 = -8x + 12 $$ 5. **Bring all terms to one side:** $$ -9x + 20 + 8x - 12 = 0 $$ $$ -x + 8 = 0 $$ 6. **Solve for $x$:** $$ -x = -8 $$ $$ x = 8 $$ 7. **Check for restrictions:** Denominators cannot be zero, so $x \neq 2$ and $x \neq 4$. Since $x=8$ is allowed, it is the solution. --- **Final answer:** $$ x = 8 $$