\2x + \frac{1}{3}y = -2\

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{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "explain_then_answer", "quick" ], "needs_drawing": true, "concepts_used": [ "Linear Intercepts", "Graphing Linear Equations" ], "new_concepts": [], "current_concepts": [ "Linear Intercepts", "Graphing Linear Equations" ] } ## Find the x-intercept $$ \begin{aligned} &y = 0 \ &2x + \frac{1}{3}(0) = -2 \ &2x = -2 \ &x = -1 \ &\text{Intercept: } (-1, 0) \end{aligned} $$ ## Find the y-intercept $$ \begin{aligned} &x = 0 \ &2(0) + \frac{1}{3}y = -2 \ &\frac{1}{3}y = -2 \ &y = -6 \ &\text{Intercept: } (0, -6) \end{aligned} $$ ## Determine the line equation in slope-intercept form $$ \begin{aligned} &\frac{1}{3}y = -2x - 2 \ &y = -6x - 6 \end{aligned} $$ To graph the linear equation $2x + \frac{1}{3}y = -2$, plot the two key intercept points and draw the line passing through them: * **$x$-intercept:** $(-1, 0)$ * **$y$-intercept:** $(0, -6)$ { "elements": [ { "type": "point", "params": [[-1, 0]], "properties": { "name": "(-1, 0)", "size": 4, "color": "#8C55F2", "withLabel": true } }, { "type": "point", "params": [[0, -6]], "properties": { "name": "(0, -6)", "size": 4, "color": "#8C55F2", "withLabel": true } }, { "type": "line", "params": [[-1, 0], [0, -6]], "properties": { "strokeColor": "#5583F2", "strokeWidth": 3 } } ], "timestamps": [0.5, 1.0, 1.5] } { "subject": "Mathematics", "question_type": "Free Response", "knowledge_point": [ "Mathematics", "Geometry", "Graphing Linear Equations" ] }

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Find the x-intercept

Find the y-intercept

Determine the line equation in slope-intercept form

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