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": [ "Number of Solutions", "Slope-Intercept Form" ], "new_concepts": [], "current_concepts": [ "Number of Solutions", "Slope-Intercept Form", "System of Equations" ] } ## Analyze system (a) $$ \begin{aligned} &y = 3x + 7 \quad (m_1 = 3, \, b_1 = 7) \ &y = 3x - 7 \quad (m_2 = 3, \, b_2 = -7) \ &m_1 = m_2 \text{ and } b_1 eq b_2 \implies 0 \text{ solutions (parallel lines)} \end{aligned} $$ ## Analyze system (b) $$ \begin{aligned} &x - 2y = -5 \implies 4(x - 2y) = 4(-5) \implies 4x - 8y = -20 \ &\text{Both equations are identical} \implies \infty \text{ solutions (coincident lines)} \end{aligned} $$ ## Analyze system (c) $$ \begin{aligned} &y = -6x + 1 \quad (m_1 = -6, \, b_1 = 1) \ &y = 6x + 1 \quad (m_2 = 6, \, b_2 = 1) \ &m_1 eq m_2 \implies 1 \text{ solution (intersecting lines)} \end{aligned} $$ a. No solution (0 solutions) b. Infinitely many solutions c. One solution (1 solution) { "subject": "Mathematics", "question_type": "Multi-part", "knowledge_point": [ "Mathematics", "Algebra", "Number of Solutions" ] }

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Analyze system (a)

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Analyze system (c)

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