Question

compute the slopes for the two given functions and determine whether the lines are parallel or perpendicular. (i)\\(y = 4x + 20\\) (ii) line in the graph

the given lines are parallel.
the given lines are perpendicular.

Explanation:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"multiple_choice_formatter",
"explain_then_answer",
"tutor",
"mastery_adaptive_brevity"
],
"needs_drawing": false,
"concepts_used": [
"Slope-Intercept Form",
"Slope of a Linear Function",
"Parallel and Perpendicular Lines"
],
"new_concepts": [],
"current_concepts": [
"Slope-Intercept Form",
"Slope of a Linear Function",
"Parallel and Perpendicular Lines"
]
}
</pre_analysis>

<reasoning>

Find slope of first line

Using the Slope-Intercept Form and Slope of a Linear Function knowledge points
\[

$$\begin{aligned} &y = 4x + 20\\ &m_1 = 4 \end{aligned}$$

\]

Find slope of second line

Using the Slope of a Linear Function knowledge point
\[

$$\begin{aligned} &(x_1, y_1) = (0, 1)\\ &(x_2, y_2) = (-1, -3)\\ &m_2 = \frac{-3 - 1}{-1 - 0} = \frac{-4}{-1} = 4 \end{aligned}$$

\]

Compare slopes

Using the Parallel and Perpendicular Lines knowledge point
\[

$$\begin{aligned} &m_1 = m_2 = 4\\ &\text{Lines are parallel.} \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-correct>The given lines are parallel.</mcq-correct>
<mcq-option>The given lines are perpendicular.</mcq-option>
</answer>

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

Answer:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"multiple_choice_formatter",
"explain_then_answer",
"tutor",
"mastery_adaptive_brevity"
],
"needs_drawing": false,
"concepts_used": [
"Slope-Intercept Form",
"Slope of a Linear Function",
"Parallel and Perpendicular Lines"
],
"new_concepts": [],
"current_concepts": [
"Slope-Intercept Form",
"Slope of a Linear Function",
"Parallel and Perpendicular Lines"
]
}
</pre_analysis>

<reasoning>

Find slope of first line

Using the Slope-Intercept Form and Slope of a Linear Function knowledge points
\[

$$\begin{aligned} &y = 4x + 20\\ &m_1 = 4 \end{aligned}$$

\]

Find slope of second line

Using the Slope of a Linear Function knowledge point
\[

$$\begin{aligned} &(x_1, y_1) = (0, 1)\\ &(x_2, y_2) = (-1, -3)\\ &m_2 = \frac{-3 - 1}{-1 - 0} = \frac{-4}{-1} = 4 \end{aligned}$$

\]

Compare slopes

Using the Parallel and Perpendicular Lines knowledge point
\[

$$\begin{aligned} &m_1 = m_2 = 4\\ &\text{Lines are parallel.} \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-correct>The given lines are parallel.</mcq-correct>
<mcq-option>The given lines are perpendicular.</mcq-option>
</answer>

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