1. The problem is to analyze the system of equations: $$x + y = 4$$ $$2x + 2y = 8$$ and understand why the student obtained the equation $0=0$ and gave the solution set as $\{(5,10), (0,2), (6, ? )\}$ (noting the student's solution seems inconsistent). 2. Start by examining the system: The second equation can be simplified by dividing both sides by 2: $$2x + 2y = 8 \implies x + y = 4$$ 3. Notice that both equations are actually the same: $$x + y = 4$$ This means the system is dependent and has infinitely many solutions along the line $x + y = 4$. 4. When solving such a system, subtracting one equation from the other yields: $$ (2x + 2y) - 2(x + y) = 8 - 2(4) \implies 0 = 0 $$ which is a true statement but does not provide new information. 5. The student's mistake is in listing specific points like $(5,10)$ and $(0,2)$ without verifying if they satisfy the equation $x + y = 4$. Check the points: - For $(5,10)$: $5 + 10 = 15 \neq 4$ - For $(0,2)$: $0 + 2 = 2 \neq 4$ - For $(6, ?)$: incomplete point, but if $y$ is chosen so that $6 + y = 4$, then $y = -2$. 6. The correct solution set is all points $(x,y)$ such that: $$x + y = 4$$ or equivalently: $$y = 4 - x$$ 7. Therefore, the student should have expressed the solution as the infinite set of points on the line $y = 4 - x$, not as isolated points that do not satisfy the equation. Final answer: The student incorrectly listed points that do not satisfy the system. The system has infinitely many solutions along the line $x + y = 4$ because the two equations are dependent, leading to the identity $0=0$ when subtracting one from the other.