1. **Stating the problem:** Solve the equation $$a - \frac{12}{b} = 4 - \frac{100}{a}$$ for $a$ given $b$ or vice versa. 2. **Understanding the equation:** The equation involves fractions with variables in denominators. We need to isolate one variable or express one in terms of the other. 3. **Rewrite the equation:** $$a - \frac{12}{b} = 4 - \frac{100}{a}$$ 4. **Bring all terms to one side:** $$a - 4 = \frac{12}{b} - \frac{100}{a}$$ 5. **Find a common denominator on the right side:** $$\frac{12}{b} - \frac{100}{a} = \frac{12a - 100b}{ab}$$ 6. **Rewrite the equation:** $$a - 4 = \frac{12a - 100b}{ab}$$ 7. **Multiply both sides by $ab$ to clear denominators:** $$ab(a - 4) = 12a - 100b$$ 8. **Expand the left side:** $$a^2b - 4ab = 12a - 100b$$ 9. **Bring all terms to one side:** $$a^2b - 4ab - 12a + 100b = 0$$ 10. **Group terms to factor:** $$a^2b - 4ab - 12a + 100b = ab(a - 4) - 12a + 100b$$ 11. **No simple factorization; solve for $a$ in terms of $b$ or vice versa. For example, solve for $a$: Rewrite as quadratic in $a$: $$a^2b - 4ab - 12a + 100b = 0$$ Divide both sides by $b$ (assuming $b \neq 0$): $$a^2 - 4a - \frac{12a}{b} + 100 = 0$$ This is complicated; better to treat original equation as is or specify values for $b$ to solve for $a$. **Alternatively, solve the original equation for $a$ explicitly:** From step 4: $$a - 4 = \frac{12}{b} - \frac{100}{a}$$ Multiply both sides by $a$: $$a(a - 4) = a \left( \frac{12}{b} - \frac{100}{a} \right)$$ $$a^2 - 4a = \frac{12a}{b} - 100$$ Bring all terms to one side: $$a^2 - 4a - \frac{12a}{b} + 100 = 0$$ Multiply both sides by $b$: $$b a^2 - 4 b a - 12 a + 100 b = 0$$ Group terms: $$a^2 b - a(4b + 12) + 100 b = 0$$ This is a quadratic in $a$: $$a^2 b - a(4b + 12) + 100 b = 0$$ 12. **Use quadratic formula for $a$:** $$a = \frac{4b + 12 \pm \sqrt{(4b + 12)^2 - 4 b \cdot 100 b}}{2b}$$ Simplify discriminant: $$(4b + 12)^2 - 400 b^2 = 16 b^2 + 96 b + 144 - 400 b^2 = -384 b^2 + 96 b + 144$$ 13. **Final solution:** $$a = \frac{4b + 12 \pm \sqrt{-384 b^2 + 96 b + 144}}{2b}$$ This expresses $a$ in terms of $b$. **Summary:** The solution for $a$ in terms of $b$ is given by the quadratic formula above.