Question

find the function which is perpendicular to (x + 2y = 6).

(2x - y = 5)
(x - y = 10)
(3x + 3y = 3)
all of the choices

Explanation:

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<reasoning>

Find the slope of the given line

\[

$$\begin{aligned} x + 2y &= 6 \\ 2y &= -x + 6 \\ y &= -\frac{1}{2}x + 3 \\ m_1 &= -\frac{1}{2} \end{aligned}$$

\]

Determine the perpendicular slope

\[

$$\begin{aligned} m_1 \cdot m_2 &= -1 \\ -\frac{1}{2} \cdot m_2 &= -1 \\ m_2 &= 2 \end{aligned}$$

\]

Identify the matching option

\[

$$\begin{aligned} 2x - y &= 5 \\ y &= 2x - 5 \quad (\text{slope } m = 2) \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-correct>(A) \(2x - y = 5\)</mcq-correct>
<mcq-option>(B) \(x - y = 10\)</mcq-option>
<mcq-option>(C) \(3x + 3y = 3\)</mcq-option>
<mcq-option>(D) All of the choices</mcq-option>
</answer>

<plot>
{
"elements": [
{
"type": "functiongraph",
"params": [{"js": "-0.5*x + 3", "latex": "y = -\frac{1}{2}x + 3"}, -10, 10],
"properties": {
"strokeColor": "#8C55F2",
"strokeWidth": 2,
"name": "x + 2y = 6",
"withLabel": true
}
},
{
"type": "functiongraph",
"params": [{"js": "2*x - 5", "latex": "y = 2x - 5"}, -10, 10],
"properties": {
"strokeColor": "#F2557F",
"strokeWidth": 2,
"name": "2x - y = 5",
"withLabel": true
}
}
],
"timestamps": [0.5, 1.0]
}
</plot>

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"Parallel and Perpendicular Lines"
]
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</post_analysis>

Answer:

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<reasoning>

Find the slope of the given line

\[

$$\begin{aligned} x + 2y &= 6 \\ 2y &= -x + 6 \\ y &= -\frac{1}{2}x + 3 \\ m_1 &= -\frac{1}{2} \end{aligned}$$

\]

Determine the perpendicular slope

\[

$$\begin{aligned} m_1 \cdot m_2 &= -1 \\ -\frac{1}{2} \cdot m_2 &= -1 \\ m_2 &= 2 \end{aligned}$$

\]

Identify the matching option

\[

$$\begin{aligned} 2x - y &= 5 \\ y &= 2x - 5 \quad (\text{slope } m = 2) \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-correct>(A) \(2x - y = 5\)</mcq-correct>
<mcq-option>(B) \(x - y = 10\)</mcq-option>
<mcq-option>(C) \(3x + 3y = 3\)</mcq-option>
<mcq-option>(D) All of the choices</mcq-option>
</answer>

<plot>
{
"elements": [
{
"type": "functiongraph",
"params": [{"js": "-0.5*x + 3", "latex": "y = -\frac{1}{2}x + 3"}, -10, 10],
"properties": {
"strokeColor": "#8C55F2",
"strokeWidth": 2,
"name": "x + 2y = 6",
"withLabel": true
}
},
{
"type": "functiongraph",
"params": [{"js": "2*x - 5", "latex": "y = 2x - 5"}, -10, 10],
"properties": {
"strokeColor": "#F2557F",
"strokeWidth": 2,
"name": "2x - y = 5",
"withLabel": true
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}
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"timestamps": [0.5, 1.0]
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</plot>

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