1. **State the problem:** We are given a system of two equations: $$x - 2y - 6 = 0$$ $$x - \frac{y}{2} + x = 4$$ We need to determine which of the given points (A, B, C, D) satisfy both equations. 2. **Rewrite the second equation for clarity:** $$x - \frac{y}{2} + x = 4 \implies 2x - \frac{y}{2} = 4$$ Multiply both sides by 2 to clear the fraction: $$\cancel{2} \times (2x - \frac{y}{2}) = \cancel{2} \times 4 \implies 4x - y = 8$$ 3. **Rewrite the first equation:** $$x - 2y - 6 = 0 \implies x - 2y = 6$$ 4. **Check each point:** **A. (2, 2)** - First equation: $2 - 2(2) = 2 - 4 = -2 \neq 6$ - So A is not a solution. **B. (-3, -2)** - First equation: $-3 - 2(-2) = -3 + 4 = 1 \neq 6$ - So B is not a solution. **C. (2, -2)** - First equation: $2 - 2(-2) = 2 + 4 = 6$ (satisfies first) - Second equation: $4(2) - (-2) = 8 + 2 = 10 \neq 8$ - So C is not a solution. **D. (\frac{2}{3}, -2)$ - First equation: $\frac{2}{3} - 2(-2) = \frac{2}{3} + 4 = \frac{2}{3} + \frac{12}{3} = \frac{14}{3} \neq 6$ - So D is not a solution. 5. **Conclusion:** None of the given points satisfy both equations exactly. 6. **Alternative: Solve the system for $(x,y)$:** From first equation: $$x = 6 + 2y$$ Substitute into second equation: $$4x - y = 8 \implies 4(6 + 2y) - y = 8$$ $$24 + 8y - y = 8$$ $$24 + 7y = 8$$ $$7y = 8 - 24 = -16$$ $$y = -\frac{16}{7}$$ Then, $$x = 6 + 2\left(-\frac{16}{7}\right) = 6 - \frac{32}{7} = \frac{42}{7} - \frac{32}{7} = \frac{10}{7}$$ So the solution is: $$\left(\frac{10}{7}, -\frac{16}{7}\right)$$ This point is not among the options A, B, C, or D. **Final answer:** None of the given points satisfy the system.