Question

if \\(3x - y = 23\\) and \\(y\\) is three times \\(x\\) increased by 19, what can be said about these two functions?

• the two lines are perpendicular.
• the two lines are parallel.

Explanation:

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

<reasoning>

Convert the first equation to slope-intercept form

Using the Linear Equations and Slope-Intercept Form knowledge points
\[

$$\begin{aligned} 3x - y &= 23 \\ -y &= -3x + 23 \\ y &= 3x - 23 \end{aligned}$$

\]

Write the second equation from the verbal description

Using the Linear Equations knowledge point
\[
y = 3x + 19
\]

Compare the slopes of the two lines

Using the Slope-Intercept Form and Parallel and Perpendicular Lines knowledge points
\[

$$\begin{aligned} m_1 &= 3 \\ m_2 &= 3 \\ m_1 &= m_2 \quad \text{and} \quad -23 e 19 \end{aligned}$$

\]
</reasoning>

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

<plot>
{
"elements": [
{
"type": "functiongraph",
"params": [{"js": "3*x - 23", "latex": "y = 3x - 23"}, -10, 15],
"properties": {
"strokeColor": "#8C55F2",
"strokeWidth": 2,
"name": "y = 3x - 23",
"withLabel": true
}
},
{
"type": "functiongraph",
"params": [{"js": "3*x + 19", "latex": "y = 3x + 19"}, -15, 10],
"properties": {
"strokeColor": "#5583F2",
"strokeWidth": 2,
"name": "y = 3x + 19",
"withLabel": true
}
}
]
}
</plot>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Algebra",
"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": true,
"concepts_used": [
"Linear Equations",
"Slope-Intercept Form",
"Parallel and Perpendicular Lines"
],
"new_concepts": [],
"current_concepts": [
"Linear Equations",
"Slope-Intercept Form",
"Parallel and Perpendicular Lines"
]
}
</pre_analysis>

<reasoning>

Convert the first equation to slope-intercept form

Using the Linear Equations and Slope-Intercept Form knowledge points
\[

$$\begin{aligned} 3x - y &= 23 \\ -y &= -3x + 23 \\ y &= 3x - 23 \end{aligned}$$

\]

Write the second equation from the verbal description

Using the Linear Equations knowledge point
\[
y = 3x + 19
\]

Compare the slopes of the two lines

Using the Slope-Intercept Form and Parallel and Perpendicular Lines knowledge points
\[

$$\begin{aligned} m_1 &= 3 \\ m_2 &= 3 \\ m_1 &= m_2 \quad \text{and} \quad -23 e 19 \end{aligned}$$

\]
</reasoning>

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

<plot>
{
"elements": [
{
"type": "functiongraph",
"params": [{"js": "3*x - 23", "latex": "y = 3x - 23"}, -10, 15],
"properties": {
"strokeColor": "#8C55F2",
"strokeWidth": 2,
"name": "y = 3x - 23",
"withLabel": true
}
},
{
"type": "functiongraph",
"params": [{"js": "3*x + 19", "latex": "y = 3x + 19"}, -15, 10],
"properties": {
"strokeColor": "#5583F2",
"strokeWidth": 2,
"name": "y = 3x + 19",
"withLabel": true
}
}
]
}
</plot>

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