Question

find the function which is perpendicular to \\(x + y = 10\\).

\\(x + y = 4\\)
\\(x - y = 10\\)
\\(x + y = 8\\)
all of the choices

Explanation:

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<reasoning>

Find the slope of the given line

\[

$$\begin{aligned} x + y &= 10 \\ y &= -x + 10 \\ m_1 &= -1 \end{aligned}$$

\]

Determine the perpendicular slope

\[

$$\begin{aligned} m_1 \cdot m_2 &= -1 \\ -1 \cdot m_2 &= -1 \\ m_2 &= 1 \end{aligned}$$

\]

Identify the matching option

\[

$$\begin{aligned} &\text{Option 1: } x + y = 4 \implies y = -x + 4 \quad (m = -1) \\ &\text{Option 2: } x - y = 10 \implies y = x - 10 \quad (m = 1) \\ &\text{Option 3: } x + y = 8 \implies y = -x + 8 \quad (m = -1) \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) \(x + y = 4\)</mcq-option>
<mcq-correct>(B) \(x - y = 10\)</mcq-correct>
<mcq-option>(C) \(x + y = 8\)</mcq-option>
<mcq-option>(D) All of the choices</mcq-option>
</answer>

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"Mathematics",
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]
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Answer:

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<reasoning>

Find the slope of the given line

\[

$$\begin{aligned} x + y &= 10 \\ y &= -x + 10 \\ m_1 &= -1 \end{aligned}$$

\]

Determine the perpendicular slope

\[

$$\begin{aligned} m_1 \cdot m_2 &= -1 \\ -1 \cdot m_2 &= -1 \\ m_2 &= 1 \end{aligned}$$

\]

Identify the matching option

\[

$$\begin{aligned} &\text{Option 1: } x + y = 4 \implies y = -x + 4 \quad (m = -1) \\ &\text{Option 2: } x - y = 10 \implies y = x - 10 \quad (m = 1) \\ &\text{Option 3: } x + y = 8 \implies y = -x + 8 \quad (m = -1) \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) \(x + y = 4\)</mcq-option>
<mcq-correct>(B) \(x - y = 10\)</mcq-correct>
<mcq-option>(C) \(x + y = 8\)</mcq-option>
<mcq-option>(D) All of the choices</mcq-option>
</answer>

<post_analysis>
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"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Geometry",
"Parallel and Perpendicular Lines"
]
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