Question

a line passes through the point \\((6, 1)\\) and has a slope of \\(\frac{3}{2}\\). write an equation for this line.

Explanation:

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"Point-Slope Form",
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"Point-Slope Form",
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<reasoning>

Identify given values

Using the Point-Slope Form knowledge point
\[
(x_1, y_1) = (6, 1),\quad m = \frac{3}{2}
\]

Apply point-slope formula

Using the Point-Slope Form knowledge point
\[
y - y_1 = m(x - x_1) \implies y - 1 = \frac{3}{2}(x - 6)
\]

Convert to slope-intercept form

Using the Slope-Intercept Form knowledge point
\[

$$\begin{aligned} y - 1 &= \frac{3}{2}x - 9\\ y &= \frac{3}{2}x - 8 \end{aligned}$$

\]
</reasoning>

<answer>
A line passes through the point \((6, 1)\) and has a slope of \(\frac{3}{2}\). Write an equation for this line.

<blank>\(y = \frac{3}{2}x - 8\)</blank>
</answer>

<plot>
{
"elements": [
{
"type": "point",
"params": [[6, 1]],
"properties": {
"name": "(6, 1)",
"size": 4,
"color": "#8C55F2",
"withLabel": true
}
},
{
"type": "point",
"params": [[0, -8]],
"properties": {
"name": "(0, -8)",
"size": 4,
"color": "#5583F2",
"withLabel": true
}
},
{
"type": "functiongraph",
"params": [{"js": "1.5*x - 8", "latex": "y = \frac{3}{2}x - 8"}, -2, 10],
"properties": {
"strokeColor": "#8C55F2",
"strokeWidth": 2
}
}
],
"timestamps": [0.5, 1.0]
}
</plot>

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{
"subject": "Mathematics",
"question_type": "Fill-in-the-blank",
"knowledge_point": [
"Mathematics",
"Algebra",
"Slope-Intercept Form"
]
}
</post_analysis>

Answer:

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"explain_then_answer",
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"concepts_used": [
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</pre_analysis>

<reasoning>

Identify given values

Using the Point-Slope Form knowledge point
\[
(x_1, y_1) = (6, 1),\quad m = \frac{3}{2}
\]

Apply point-slope formula

Using the Point-Slope Form knowledge point
\[
y - y_1 = m(x - x_1) \implies y - 1 = \frac{3}{2}(x - 6)
\]

Convert to slope-intercept form

Using the Slope-Intercept Form knowledge point
\[

$$\begin{aligned} y - 1 &= \frac{3}{2}x - 9\\ y &= \frac{3}{2}x - 8 \end{aligned}$$

\]
</reasoning>

<answer>
A line passes through the point \((6, 1)\) and has a slope of \(\frac{3}{2}\). Write an equation for this line.

<blank>\(y = \frac{3}{2}x - 8\)</blank>
</answer>

<plot>
{
"elements": [
{
"type": "point",
"params": [[6, 1]],
"properties": {
"name": "(6, 1)",
"size": 4,
"color": "#8C55F2",
"withLabel": true
}
},
{
"type": "point",
"params": [[0, -8]],
"properties": {
"name": "(0, -8)",
"size": 4,
"color": "#5583F2",
"withLabel": true
}
},
{
"type": "functiongraph",
"params": [{"js": "1.5*x - 8", "latex": "y = \frac{3}{2}x - 8"}, -2, 10],
"properties": {
"strokeColor": "#8C55F2",
"strokeWidth": 2
}
}
],
"timestamps": [0.5, 1.0]
}
</plot>

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"subject": "Mathematics",
"question_type": "Fill-in-the-blank",
"knowledge_point": [
"Mathematics",
"Algebra",
"Slope-Intercept Form"
]
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