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{ "quality": "blurry", "question_count": 3, "skills_matched": [ "step_cot", "explain_then_answer", "quick" ], "needs_drawing": true, "concepts_used": [ "Linear Equations" ], "new_concepts": [ "Systems of Linear Equations", "Linear Modeling" ], "current_concepts": [ "Linear Equations", "Systems of Linear Equations", "Linear Modeling" ] } ## Solve the system of linear equations graphically $$ \begin{aligned} &y = 3x - 5 \ &2x + 5y = 10 \implies 5y = -2x + 10 \implies y = -0.4x + 2 \ &3x - 5 = -0.4x + 2 \implies 3.4x = 7 \implies x = \frac{35}{17} \approx 2.06 \ &y = 3(2.06) - 5 \approx 1.18 \ &\text{Approximate intersection point: } (2.1, 1.2) \end{aligned} $$ ## Determine the equation for the reading graph $$ \begin{aligned} &\text{y-intercept } (0, 15) \implies b = 15 \ &\text{Point on line: } (4, 5) \ &m = \frac{5 - 15}{4 - 0} = -2.5 \ &y = -2.5x + 15 \end{aligned} $$ ## Determine the equation for the banquet cost table $$ \begin{aligned} &\text{Points: } (135, 8250), (180, 10950) \ &m = \frac{10950 - 8250}{180 - 135} = \frac{2700}{45} = 60 \ &C - 8250 = 60(P - 135) \ &C = 60P - 8100 + 8250 \implies C = 60P + 150 \end{aligned} $$ ## Question 8 The approximate point of intersection is $(2.1, 1.2)$. ## Question 9 The equation for the graph is $y = -2.5x + 15$ (where $y$ is the chapters left and $x$ is the number of evenings reading). ## Question 10 The equation for the table is $C = 60P + 150$ (where $C$ is the cost of the banquet and $P$ is the number of people). { "elements": [ { "type": "functiongraph", "params": [ { "js": "3*x - 5", "latex": "y = 3x - 5" }, -2, 6 ], "properties": { "strokeColor": "#8C55F2", "strokeWidth": 2, "name": "y = 3x - 5", "withLabel": true } }, { "type": "functiongraph", "params": [ { "js": "-0.4*x + 2", "latex": "2x + 5y = 10" }, -2, 6 ], "properties": { "strokeColor": "#5583F2", "strokeWidth": 2, "name": "2x + 5y = 10", "withLabel": true } }, { "type": "point", "params": [ [ 2.0588, 1.1765 ] ], "properties": { "name": "(2.1, 1.2)", "size": 4, "color": "#F2557F", "withLabel": true } } ], "timestamps": [ 0.5, 1.0, 1.5 ] } { "subject": "Mathematics", "question_type": "Multi-part", "knowledge_point": [ "Mathematics", "Algebra", "Systems of Linear Equations" ] }

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Solve the system of linear equations graphically

Determine the equation for the reading graph

Determine the equation for the banquet cost table

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Question 8

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Question 10