1. **Problem statement:** Solve the system of equations using the Gleichsetzungsverfahren (equalization method) for part (a): (I) $4x + 5y = 12$ (II) $10x + 2y = 9$ 2. **Formula and method:** The Gleichsetzungsverfahren involves solving both equations for the same variable and then setting these expressions equal to each other. 3. **Step 1: Solve (I) for $y$:** $$4x + 5y = 12 \Rightarrow 5y = 12 - 4x \Rightarrow y = \frac{12 - 4x}{5}$$ 4. **Step 2: Solve (II) for $y$:** $$10x + 2y = 9 \Rightarrow 2y = 9 - 10x \Rightarrow y = \frac{9 - 10x}{2}$$ 5. **Step 3: Set the two expressions for $y$ equal:** $$\frac{12 - 4x}{5} = \frac{9 - 10x}{2}$$ 6. **Step 4: Cross-multiply to solve for $x$:** $$2(12 - 4x) = 5(9 - 10x)$$ $$24 - 8x = 45 - 50x$$ 7. **Step 5: Rearrange terms:** $$24 - 8x + 50x = 45$$ $$24 + 42x = 45$$ 8. **Step 6: Isolate $x$:** $$42x = 45 - 24$$ $$42x = 21$$ $$x = \frac{21}{42} = \frac{\cancel{21}}{\cancel{42}} = \frac{1}{2}$$ 9. **Step 7: Substitute $x = \frac{1}{2}$ into (I) to find $y$:** $$4\left(\frac{1}{2}\right) + 5y = 12$$ $$2 + 5y = 12$$ $$5y = 12 - 2 = 10$$ $$y = \frac{10}{5} = 2$$ 10. **Final solution:** $$\boxed{x = \frac{1}{2}, y = 2}$$ This completes the solution for the first system using the Gleichsetzungsverfahren.