Question

question 16 points 2

choose the parallel lines from the following lines.

\\(y = 6x + 2\\) and \\(3x - 2y = 4\\)

\\(y = 5x + 2\\) and \\(5x - y = 2\\)

Explanation:

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"Parallel and Perpendicular Lines"
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<reasoning>

Find the slopes of the first pair of lines

\[

$$\begin{aligned} &\text{Line 1: } y = 6x + 2 \implies m_1 = 6\\ &\text{Line 2: } 3x - 2y = 4 \implies 2y = 3x - 4 \implies y = \frac{3}{2}x - 2 \implies m_2 = \frac{3}{2}\\ &m_1 eq m_2 \end{aligned}$$

\]

Find the slopes of the second pair of lines

\[

$$\begin{aligned} &\text{Line 1: } y = 5x + 2 \implies m_1 = 5\\ &\text{Line 2: } 5x - y = 2 \implies y = 5x - 2 \implies m_2 = 5\\ &m_1 = m_2 = 5 \end{aligned}$$

\]

Identify the parallel lines

\[

$$\begin{aligned} &\text{Since the second pair of lines have equal slopes } (m = 5) \text{ and different } y\text{-intercepts } (2 eq -2)\text{, they are parallel.} \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) \(y = 6x + 2\) and \(3x - 2y = 4\)</mcq-option>
<mcq-correct>(B) \(y = 5x + 2\) and \(5x - y = 2\)</mcq-correct>
</answer>

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"question_type": "Multiple Choice",
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"Parallel and Perpendicular Lines"
]
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</post_analysis>

Answer:

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<reasoning>

Find the slopes of the first pair of lines

\[

$$\begin{aligned} &\text{Line 1: } y = 6x + 2 \implies m_1 = 6\\ &\text{Line 2: } 3x - 2y = 4 \implies 2y = 3x - 4 \implies y = \frac{3}{2}x - 2 \implies m_2 = \frac{3}{2}\\ &m_1 eq m_2 \end{aligned}$$

\]

Find the slopes of the second pair of lines

\[

$$\begin{aligned} &\text{Line 1: } y = 5x + 2 \implies m_1 = 5\\ &\text{Line 2: } 5x - y = 2 \implies y = 5x - 2 \implies m_2 = 5\\ &m_1 = m_2 = 5 \end{aligned}$$

\]

Identify the parallel lines

\[

$$\begin{aligned} &\text{Since the second pair of lines have equal slopes } (m = 5) \text{ and different } y\text{-intercepts } (2 eq -2)\text{, they are parallel.} \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) \(y = 6x + 2\) and \(3x - 2y = 4\)</mcq-option>
<mcq-correct>(B) \(y = 5x + 2\) and \(5x - y = 2\)</mcq-correct>
</answer>

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