Question

what is the equation, in point-slope form, of the line that is parallel to the given line and passes through the point \\((-3, 1)\\)?

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

Explanation:

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<reasoning>

Determine the slope of the given line

\[

$$\begin{aligned} (x_1, y_1) &= (-2, -4) \\ (x_2, y_2) &= (2, 2) \\ m &= \frac{2 - (-4)}{2 - (-2)} = \frac{6}{4} = \frac{3}{2} \end{aligned}$$

\]

Identify the slope of the parallel line

\[
m_{\text{parallel}} = m = \frac{3}{2}
\]

Write the equation in point-slope form

\[

$$\begin{aligned} (x_0, y_0) &= (-3, 1) \\ y - y_0 &= m_{\text{parallel}}(x - x_0) \\ y - 1 &= \frac{3}{2}(x - (-3)) \\ y - 1 &= \frac{3}{2}(x + 3) \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) \(y - 1 = -\frac{3}{2}(x + 3)\)</mcq-option>
<mcq-option>(B) \(y - 1 = -\frac{2}{3}(x + 3)\)</mcq-option>
<mcq-option>(C) \(y - 1 = \frac{2}{3}(x + 3)\)</mcq-option>
<mcq-correct>(D) \(y - 1 = \frac{3}{2}(x + 3)\)</mcq-correct>
</answer>

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"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Geometry",
"Point-Slope Form"
]
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</post_analysis>

Answer:

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<reasoning>

Determine the slope of the given line

\[

$$\begin{aligned} (x_1, y_1) &= (-2, -4) \\ (x_2, y_2) &= (2, 2) \\ m &= \frac{2 - (-4)}{2 - (-2)} = \frac{6}{4} = \frac{3}{2} \end{aligned}$$

\]

Identify the slope of the parallel line

\[
m_{\text{parallel}} = m = \frac{3}{2}
\]

Write the equation in point-slope form

\[

$$\begin{aligned} (x_0, y_0) &= (-3, 1) \\ y - y_0 &= m_{\text{parallel}}(x - x_0) \\ y - 1 &= \frac{3}{2}(x - (-3)) \\ y - 1 &= \frac{3}{2}(x + 3) \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) \(y - 1 = -\frac{3}{2}(x + 3)\)</mcq-option>
<mcq-option>(B) \(y - 1 = -\frac{2}{3}(x + 3)\)</mcq-option>
<mcq-option>(C) \(y - 1 = \frac{2}{3}(x + 3)\)</mcq-option>
<mcq-correct>(D) \(y - 1 = \frac{3}{2}(x + 3)\)</mcq-correct>
</answer>

<post_analysis>
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"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Geometry",
"Point-Slope Form"
]
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