1. **State the problem:** We need to find which test point among (2,1), (3,0), (1,2), and (0,0) satisfies the system of inequalities: $$\begin{cases} 10y - 5x \leq 0 \\ 4x + 2y > 10 \end{cases}$$ 2. **Rewrite inequalities for clarity:** - First inequality: $$10y - 5x \leq 0$$ can be simplified by dividing both sides by 5: $$\cancel{5} \times 2y - \cancel{5} \times x \leq 0 \Rightarrow 2y - x \leq 0$$ - Second inequality: $$4x + 2y > 10$$ can be simplified by dividing both sides by 2: $$\cancel{2} \times 2x + \cancel{2} \times y > 5 \Rightarrow 2x + y > 5$$ 3. **Check each test point:** - For (2,1): - Check $$2y - x \leq 0$$: $$2(1) - 2 = 2 - 2 = 0 \leq 0$$ (True) - Check $$2x + y > 5$$: $$2(2) + 1 = 4 + 1 = 5 > 5$$? No, 5 is not greater than 5 (False) - For (3,0): - Check $$2y - x \leq 0$$: $$2(0) - 3 = 0 - 3 = -3 \leq 0$$ (True) - Check $$2x + y > 5$$: $$2(3) + 0 = 6 + 0 = 6 > 5$$ (True) - For (1,2): - Check $$2y - x \leq 0$$: $$2(2) - 1 = 4 - 1 = 3 \leq 0$$? No (False) - For (0,0): - Check $$2y - x \leq 0$$: $$2(0) - 0 = 0 \leq 0$$ (True) - Check $$2x + y > 5$$: $$2(0) + 0 = 0 > 5$$? No (False) 4. **Conclusion:** Only point (3,0) satisfies both inequalities. **Final answer:** (3,0)