6) determine the number of solutions for each system of equations. justify your answers.
a. $y = 3x + 7$ and $y = 3x - 7$
b. $x - 2y = -5$ and $4x - 8y = -20$
c. $y = -6x + 1$ and $y = 6x + 1$

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{ "quality": "clear", "question_count": 3, "skills_matched": [ "step_cot", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [], "new_concepts": [ "System of Equations", "Number of Solutions" ], "current_concepts": [ "System of Equations", "Number of Solutions", "Linear Equations" ] } ## Analyze system a $$ \begin{aligned} &y = 3x + 7 \ &y = 3x - 7 \ &m_1 = m_2 = 3, \quad b_1 = 7 eq b_2 = -7 \ &\text{Parallel lines} \implies 0 \text{ solutions} \end{aligned} $$ ## Analyze system b $$ \begin{aligned} &x - 2y = -5 \implies 4x - 8y = -20 \ &4x - 8y = -20 \ &\text{Identical lines} \implies \infty \text{ solutions} \end{aligned} $$ ## Analyze system c $$ \begin{aligned} &y = -6x + 1 \ &y = 6x + 1 \ &m_1 = -6 eq m_2 = 6 \ &\text{Intersecting lines} \implies 1 \text{ solution} \end{aligned} $$ ## Question 6a 0 solutions ## Question 6b Infinite solutions ## Question 6c 1 solution { "subject": "Mathematics", "question_type": "Multi-part", "knowledge_point": [ "Mathematics", "Algebra", "System of Equations" ] }

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Question 6c