7. if $x_1 = 10$, $x_2 = 2$, and $y_1 = 85$, determine the value of $y_2$.
8. if $x_1 = 45$, $y_1 = 0.2$, and $y_2 = 9$, determine the value of $x_2$.

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{ "quality": "clear", "question_count": 2, "skills_matched": [ "step_cot", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Proportional Reasoning", "Inverse Proportion Equations", "Direct Proportion Equations" ], "new_concepts": [ "Joint Variation", "Combined Variation" ], "current_concepts": [ "Proportional Reasoning", "Inverse Proportion Equations", "Direct Proportion Equations", "Joint Variation" ] } ## Identify the variation relationship $$ \begin{aligned} &\text{Based on typical textbook problems in this section: } y \text{ varies inversely as } x \ &y = \frac{k}{x} \implies k = x \cdot y \end{aligned} $$ ## Solve Question 7 $$ \begin{aligned} &k = x_2 \cdot y_2 = 2 \cdot 85 = 170 \ &y_1 = \frac{k}{x_1} = \frac{170}{10} = 17 \end{aligned} $$ ## Solve Question 8 $$ \begin{aligned} &k = x_1 \cdot y_1 = 45 \cdot 0.2 = 9 \ &x_2 = \frac{k}{y_2} = \frac{9}{9} = 1 \end{aligned} $$ ## Question 7 $y_1 = 17$ ## Question 8 $x_2 = 1$ { "subject": "Mathematics", "question_type": "Multi-part", "knowledge_point": [ "Mathematics", "Algebra", "Inverse Proportion Equations" ] }

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Identify the variation relationship

Solve Question 7

Solve Question 8

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Question 7

Question 8