Question

find the function which is parallel to \\(3x - 27 = 3y\\).

\\(4x + 5y = 9\\)

\\(4x - 4y = 8\\)

Explanation:

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

<reasoning>

Find the slope of the given line

\[

$$\begin{aligned} 3x - 27 &= 3y \\ y &= x - 9 \end{aligned}$$

\]
The slope of the given line is \(m = 1\).

Determine the slope of the parallel line

Parallel lines have equal slopes. Therefore, the parallel line must also have a slope of \(m = 1\).

Test the given options

For the first option:
\[

$$\begin{aligned} 4x + 5y &= 9 \\ 5y &= -4x + 9 \\ y &= -\frac{4}{5}x + \frac{9}{5} \quad (\text{slope } m = -\frac{4}{5}) \end{aligned}$$

\]

For the second option:
\[

$$\begin{aligned} 4x - 4y &= 8 \\ -4y &= -4x + 8 \\ y &= x - 2 \quad (\text{slope } m = 1) \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) \(4x + 5y = 9\)</mcq-option>
<mcq-correct>(B) \(4x - 4y = 8\)</mcq-correct>
</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",
"quick"
],
"needs_drawing": false,
"concepts_used": [],
"new_concepts": [],
"current_concepts": [
"Slope of a Linear Function",
"Parallel and Perpendicular Lines"
]
}
</pre_analysis>

<reasoning>

Find the slope of the given line

\[

$$\begin{aligned} 3x - 27 &= 3y \\ y &= x - 9 \end{aligned}$$

\]
The slope of the given line is \(m = 1\).

Determine the slope of the parallel line

Parallel lines have equal slopes. Therefore, the parallel line must also have a slope of \(m = 1\).

Test the given options

For the first option:
\[

$$\begin{aligned} 4x + 5y &= 9 \\ 5y &= -4x + 9 \\ y &= -\frac{4}{5}x + \frac{9}{5} \quad (\text{slope } m = -\frac{4}{5}) \end{aligned}$$

\]

For the second option:
\[

$$\begin{aligned} 4x - 4y &= 8 \\ -4y &= -4x + 8 \\ y &= x - 2 \quad (\text{slope } m = 1) \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) \(4x + 5y = 9\)</mcq-option>
<mcq-correct>(B) \(4x - 4y = 8\)</mcq-correct>
</answer>

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