1. **State the problem:** We want to find when the function $f(y) = 10^{6} - 2 \times 10^{4} \times y$ equals $10^{5} + 5 \times 10^{3} \times x$. 2. **Set the two expressions equal:** $$10^{6} - 2 \times 10^{4} y = 10^{5} + 5 \times 10^{3} x$$ 3. **Isolate terms to express $x$ in terms of $y$:** Subtract $10^{5}$ from both sides: $$10^{6} - 10^{5} - 2 \times 10^{4} y = 5 \times 10^{3} x$$ 4. **Simplify constants:** $$10^{6} - 10^{5} = 1,000,000 - 100,000 = 900,000$$ So, $$900,000 - 20,000 y = 5,000 x$$ 5. **Solve for $x$:** $$x = \frac{900,000 - 20,000 y}{5,000}$$ 6. **Simplify the fraction by dividing numerator and denominator by 1,000:** $$x = \frac{\cancel{900,000} / 1,000 - \cancel{20,000} / 1,000 y}{\cancel{5,000} / 1,000} = \frac{900 - 20 y}{5}$$ 7. **Divide numerator terms by 5:** $$x = \frac{900}{5} - \frac{20 y}{5} = 180 - 4 y$$ **Final answer:** $$\boxed{x = 180 - 4 y}$$ This means for any value of $y$, $x$ must be $180 - 4 y$ for the two expressions to be equal.