Question

6 andrew bought 3 apple pies and 2 chocolate pies for $14.50. for $13 more, kyra bought 5 apple pies and 4 chocolate pies. which system of equations can be used to find the cost of an apple pie, a, and the cost of a chocolate pie, c?

f \\(\

$$\begin{cases} 3a + 2c = 14.50 \\\\ 5a + 4c = 27.50 \\end{cases}$$

\\)

g \\(\

$$\begin{cases} 3a + 2c = 14.50 \\\\ 5a + 4c = 13 \\end{cases}$$

\\)

h \\(\

$$\begin{cases} 2a + 3c = 14.50 \\\\ 4a + 5c = 27.50 \\end{cases}$$

\\)

j \\(\

$$\begin{cases} 2a + 3c = 13 \\\\ 4a + 5c = 14.50 \\end{cases}$$

\\)

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": [
"Systems of Linear Equations",
"Linear Function Modeling"
],
"new_concepts": [],
"current_concepts": [
"Systems of Linear Equations",
"Linear Function Modeling"
]
}
</pre_analysis>

<reasoning>

Define variables and model Andrew's purchase

Using the Linear Function Modeling knowledge point
\[

$$\begin{aligned} &a = \text{cost of an apple pie}\\ &c = \text{cost of a chocolate pie}\\ &3a + 2c = 14.50 \end{aligned}$$

\]

Model Kyra's purchase

Using the Linear Function Modeling knowledge point
\[

$$\begin{aligned} &\text{Kyra's cost} = 14.50 + 13 = 27.50\\ &5a + 4c = 27.50 \end{aligned}$$

\]

Formulate the system of equations

Using the Systems of Linear Equations knowledge point
\[

$$\begin{cases} 3a + 2c = 14.50\\ 5a + 4c = 27.50 \end{cases}$$

\]

Verify the correct option

Using the Systems of Linear Equations knowledge point
\[

$$\begin{aligned} &\text{Option F matches the system exactly.}\\ &\text{Option G incorrectly sets Kyra's total to 13.}\\ &\text{Options H and J swap the coefficients of } a \text{ and } c. \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-correct>F \(

$$\begin{cases} 3a + 2c = 14.50 \\ 5a + 4c = 27.50 \end{cases}$$

\)</mcq-correct>
<mcq-option>G \(

$$\begin{cases} 3a + 2c = 14.50 \\ 5a + 4c = 13 \end{cases}$$

\)</mcq-option>
<mcq-option>H \(

$$\begin{cases} 2a + 3c = 14.50 \\ 4a + 5c = 27.50 \end{cases}$$

\)</mcq-option>
<mcq-option>J \(

$$\begin{cases} 2a + 3c = 13 \\ 4a + 5c = 14.50 \end{cases}$$

\)</mcq-option>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Algebra",
"Systems of Linear Equations"
]
}
</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": [
"Systems of Linear Equations",
"Linear Function Modeling"
],
"new_concepts": [],
"current_concepts": [
"Systems of Linear Equations",
"Linear Function Modeling"
]
}
</pre_analysis>

<reasoning>

Define variables and model Andrew's purchase

Using the Linear Function Modeling knowledge point
\[

$$\begin{aligned} &a = \text{cost of an apple pie}\\ &c = \text{cost of a chocolate pie}\\ &3a + 2c = 14.50 \end{aligned}$$

\]

Model Kyra's purchase

Using the Linear Function Modeling knowledge point
\[

$$\begin{aligned} &\text{Kyra's cost} = 14.50 + 13 = 27.50\\ &5a + 4c = 27.50 \end{aligned}$$

\]

Formulate the system of equations

Using the Systems of Linear Equations knowledge point
\[

$$\begin{cases} 3a + 2c = 14.50\\ 5a + 4c = 27.50 \end{cases}$$

\]

Verify the correct option

Using the Systems of Linear Equations knowledge point
\[

$$\begin{aligned} &\text{Option F matches the system exactly.}\\ &\text{Option G incorrectly sets Kyra's total to 13.}\\ &\text{Options H and J swap the coefficients of } a \text{ and } c. \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-correct>F \(

$$\begin{cases} 3a + 2c = 14.50 \\ 5a + 4c = 27.50 \end{cases}$$

\)</mcq-correct>
<mcq-option>G \(

$$\begin{cases} 3a + 2c = 14.50 \\ 5a + 4c = 13 \end{cases}$$

\)</mcq-option>
<mcq-option>H \(

$$\begin{cases} 2a + 3c = 14.50 \\ 4a + 5c = 27.50 \end{cases}$$

\)</mcq-option>
<mcq-option>J \(

$$\begin{cases} 2a + 3c = 13 \\ 4a + 5c = 14.50 \end{cases}$$

\)</mcq-option>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Algebra",
"Systems of Linear Equations"
]
}
</post_analysis>