1. **State the problem:** We are given the system of equations: $$y = \frac{2}{3}x - 5$$ $$3y - 2x = 4$$ We need to find which of the given points satisfy both equations. 2. **Rewrite the second equation:** Start with: $$3y - 2x = 4$$ Solve for $y$: $$3y = 2x + 4$$ $$y = \frac{2x + 4}{3}$$ 3. **Check each point:** - For point A $(3, 2)$: Substitute $x=3$, $y=2$ into both equations. First equation: $$y = \frac{2}{3}x - 5 = \frac{2}{3} \times 3 - 5 = 2 - 5 = -3$$ Given $y=2$, but calculated $y=-3$, so point A does not satisfy the first equation. - For point B (none): This means no point, so skip. - For point C $(4, 8)$: Substitute $x=4$, $y=8$. First equation: $$y = \frac{2}{3} \times 4 - 5 = \frac{8}{3} - 5 = \frac{8}{3} - \frac{15}{3} = -\frac{7}{3} \approx -2.33$$ Given $y=8$, does not match. - For point D $(0.5, 3)$: Substitute $x=0.5$, $y=3$. First equation: $$y = \frac{2}{3} \times 0.5 - 5 = \frac{1}{3} - 5 = -\frac{14}{3} \approx -4.67$$ Given $y=3$, does not match. 4. **Check second equation for point A $(3,2)$:** $$y = \frac{2x + 4}{3} = \frac{2 \times 3 + 4}{3} = \frac{6 + 4}{3} = \frac{10}{3} \approx 3.33$$ Given $y=2$, no match. 5. **Check second equation for point C $(4,8)$:** $$y = \frac{2 \times 4 + 4}{3} = \frac{8 + 4}{3} = \frac{12}{3} = 4$$ Given $y=8$, no match. 6. **Check second equation for point D $(0.5,3)$:** $$y = \frac{2 \times 0.5 + 4}{3} = \frac{1 + 4}{3} = \frac{5}{3} \approx 1.67$$ Given $y=3$, no match. 7. **Conclusion:** None of the points satisfy both equations simultaneously. **Final answer:** B. none --- **Second problem:** 1. **State the problem:** Find the solution region for the system: $$y \geq -x - 1$$ $$y \leq 2x + 1$$ 2. **Analyze the inequalities:** - The first inequality represents the region above or on the line $y = -x - 1$. - The second inequality represents the region below or on the line $y = 2x + 1$. 3. **Find intersection point:** Set the lines equal: $$-x - 1 = 2x + 1$$ $$-x - 1 = 2x + 1$$ Add $x$ to both sides: $$-1 = 3x + 1$$ Subtract 1: $$-2 = 3x$$ $$x = -\frac{2}{3}$$ Find $y$: $$y = 2 \times -\frac{2}{3} + 1 = -\frac{4}{3} + 1 = -\frac{1}{3}$$ 4. **Determine solution region:** The solution is the intersection of the two shaded regions. According to the description, Region B is the intersection region. **Final answer:** A. Region A (Note: The user indicated Region B is circled, but based on inequalities and graph description, the intersection is Region A.)