Question

what is the equation of the line that is parallel to the given line and passes through the point \\((2, 3)\\)?

\\(x + 2y = 4\\)
\\(x + 2y = 8\\)
\\(2x + y = 4\\)
\\(2x + y = 8\\)

Explanation:

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"concepts_used": [],
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"current_concepts": [
"Slope of a Line",
"Parallel Line Equations",
"Point-Slope Form"
]
}
</pre_analysis>

<reasoning>

Find the slope of the given line

The given line passes through the points \((-4, 0)\) and \((4, -4)\).
\[
m = \frac{-4 - 0}{4 - (-4)} = \frac{-4}{8} = -\frac{1}{2}
\]

Determine the equation of the parallel line

A parallel line has the same slope, \(m = -\frac{1}{2}\). Using the point-slope form with the point \((2, 3)\):
\[

$$\begin{aligned} y - 3 &= -\frac{1}{2}(x - 2) \\ 2(y - 3) &= -(x - 2) \\ 2y - 6 &= -x + 2 \\ x + 2y &= 8 \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) \(x + 2y = 4\)</mcq-option>
<mcq-correct>(B) \(x + 2y = 8\)</mcq-correct>
<mcq-option>(C) \(2x + y = 4\)</mcq-option>
<mcq-option>(D) \(2x + y = 8\)</mcq-option>
</answer>

<plot>
{
"elements": [
{
"type": "point",
"params": [[-4, 0]],
"properties": {
"name": "(-4,0)",
"color": "#583C87",
"withLabel": true
}
},
{
"type": "point",
"params": [[4, -4]],
"properties": {
"name": "(4,-4)",
"color": "#583C87",
"withLabel": true
}
},
{
"type": "line",
"params": [[-4, 0], [4, -4]],
"properties": {
"strokeColor": "#5583F2",
"strokeWidth": 2
}
},
{
"type": "point",
"params": [[2, 3]],
"properties": {
"name": "(2,3)",
"color": "#8C55F2",
"withLabel": true
}
},
{
"type": "line",
"params": [[2, 3], [0, 4]],
"properties": {
"strokeColor": "#8C55F2",
"strokeWidth": 2,
"dash": 2
}
}
]
}
</plot>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Geometry",
"Parallel Line Equations"
]
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</post_analysis>

Answer:

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<reasoning>

Find the slope of the given line

The given line passes through the points \((-4, 0)\) and \((4, -4)\).
\[
m = \frac{-4 - 0}{4 - (-4)} = \frac{-4}{8} = -\frac{1}{2}
\]

Determine the equation of the parallel line

A parallel line has the same slope, \(m = -\frac{1}{2}\). Using the point-slope form with the point \((2, 3)\):
\[

$$\begin{aligned} y - 3 &= -\frac{1}{2}(x - 2) \\ 2(y - 3) &= -(x - 2) \\ 2y - 6 &= -x + 2 \\ x + 2y &= 8 \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) \(x + 2y = 4\)</mcq-option>
<mcq-correct>(B) \(x + 2y = 8\)</mcq-correct>
<mcq-option>(C) \(2x + y = 4\)</mcq-option>
<mcq-option>(D) \(2x + y = 8\)</mcq-option>
</answer>

<plot>
{
"elements": [
{
"type": "point",
"params": [[-4, 0]],
"properties": {
"name": "(-4,0)",
"color": "#583C87",
"withLabel": true
}
},
{
"type": "point",
"params": [[4, -4]],
"properties": {
"name": "(4,-4)",
"color": "#583C87",
"withLabel": true
}
},
{
"type": "line",
"params": [[-4, 0], [4, -4]],
"properties": {
"strokeColor": "#5583F2",
"strokeWidth": 2
}
},
{
"type": "point",
"params": [[2, 3]],
"properties": {
"name": "(2,3)",
"color": "#8C55F2",
"withLabel": true
}
},
{
"type": "line",
"params": [[2, 3], [0, 4]],
"properties": {
"strokeColor": "#8C55F2",
"strokeWidth": 2,
"dash": 2
}
}
]
}
</plot>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Geometry",
"Parallel Line Equations"
]
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