Determine which of the following points are included in the solution set to the following system of linear inequalities. \[ \begin{aligned} x+3 y & >-8 \\ -2 x+3 y & \leq 1 \end{aligned} \] Answer \( (8,4) \) \( (0,0) \) \( (-8,-2) \) \( \left(-7,-\frac{1}{3}\right) \) \( (7,5) \)

To determine which points satisfy the system of linear inequalities: \[ \begin{aligned} x + 3y &> -8 \\ -2x + 3y &\leq 1 \end{aligned} \] we will evaluate each point individually. ### 1. Point \( (8, 4) \) **First Inequality:** \[ x + 3y = 8 + 3(4) = 8 + 12 = 20 > -8 \quad \text{(True)} \] **Second Inequality:** \[ -2x + 3y = -2(8) + 3(4) = -16 + 12 = -4 \leq 1 \quad \text{(True)} \] **Conclusion:** \( (8, 4) \) **is in the solution set**. --- ### 2. Point \( (0, 0) \) **First Inequality:** \[ x + 3y = 0 + 3(0) = 0 > -8 \quad \text{(True)} \] **Second Inequality:** \[ -2x + 3y = -2(0) + 3(0) = 0 \leq 1 \quad \text{(True)} \] **Conclusion:** \( (0, 0) \) **is in the solution set**. --- ### 3. Point \( (-8, -2) \) **First Inequality:** \[ x + 3y = -8 + 3(-2) = -8 - 6 = -14 > -8 \quad \text{(False)} \] **Conclusion:** \( (-8, -2) \) **is \textbf{not} in the solution set**. --- ### 4. Point \( \left(-7, -\frac{1}{3}\right) \) **First Inequality:** \[ x + 3y = -7 + 3\left(-\frac{1}{3}\right) = -7 - 1 = -8 > -8 \quad \text{(False)} \] **Second Inequality:** \[ -2x + 3y = -2(-7) + 3\left(-\frac{1}{3}\right) = 14 - 1 = 13 \leq 1 \quad \text{(False)} \] **Conclusion:** \( \left(-7, -\frac{1}{3}\right) \) **is \textbf{not} in the solution set**. --- ### 5. Point \( (7, 5) \) **First Inequality:** \[ x + 3y = 7 + 3(5) = 7 + 15 = 22 > -8 \quad \text{(True)} \] **Second Inequality:** \[ -2x + 3y = -2(7) + 3(5) = -14 + 15 = 1 \leq 1 \quad \text{(True)} \] **Conclusion:** \( (7, 5) \) **is in the solution set**. --- **Final Answer:** All points except \( (-8, -2) \) and \( \left(-7, -\frac{1}{3}\right) \) satisfy the system. Thus, the solution set includes: - \( (8, 4) \) - \( (0, 0) \) - \( (7, 5) \) **Answer:** All points except (–8, –2) and (–7, –1⁄3) satisfy both inequalities. The solution set includes  (8, 4),  (0, 0), and  (7, 5).