1. **State the problem:** Find the values of $x$ and $y$ such that $$x^2 - 7x + 9iy = iy + 20i - 12.$$ 2. **Rewrite the equation:** Group real and imaginary parts separately: $$x^2 - 7x + 9iy = iy + 20i - 12.$$ 3. **Separate real and imaginary parts:** Real part: $$x^2 - 7x = -12.$$ Imaginary part: $$9iy = iy + 20i.$$ 4. **Solve the imaginary part:** Divide both sides by $i$ (assuming $i \neq 0$): $$9y = y + 20.$$ Simplify: $$9y - y = 20 \implies 8y = 20 \implies y = \frac{20}{8} = 2.5.$$ 5. **Solve the real part:** $$x^2 - 7x = -12 \implies x^2 - 7x + 12 = 0.$$ Factor the quadratic: $$(x - 3)(x - 4) = 0.$$ So, $$x = 3$$ or $$x = 4.$$ 6. **Final answer:** $$y = 2.5,$$ $$x = 3 \text{ or } x = 4.$$