Question

select the correct answer.

sam is an artist, and he wants to purchase frames to display his work at home. he wants to frame no fewer than 10 of his pieces and spend no more than $225. large frames cost $24, and medium frames cost $18.

which system of inequalities can sam use to determine the number of large frames, \\(x\\), and medium frames, \\(y\\), that he can purchase?

\\(\

$$\begin{aligned} &18x + 24y \\le 225 \\\\ &x + y \\ge 10 \\end{aligned}$$

\\)

\\(\

$$\begin{aligned} &24x + 18y \\ge 225 \\\\ &x + y \\le 10 \\end{aligned}$$

\\)

\\(\

$$\begin{aligned} &18x + 24y \\ge 225 \\\\ &x + y \\le 10 \\end{aligned}$$

\\)

\\(\

$$\begin{aligned} &24x + 18y \\le 225 \\\\ &x + y \\ge 10 \\end{aligned}$$

\\)

Explanation:

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<reasoning>

Define variables and quantity constraint

Using the Systems of Linear Inequalities knowledge point
\[

$$\begin{aligned} &x \ge 0, \quad y \ge 0 \\ &x + y \ge 10 \end{aligned}$$

\]

Define cost constraint

Using the Systems of Linear Inequalities knowledge point
\[

$$\begin{aligned} &\text{Cost of large frames: } 24x \\ &\text{Cost of medium frames: } 18y \\ &24x + 18y \le 225 \end{aligned}$$

\]

Combine into system of inequalities

Using the Systems of Linear Inequalities knowledge point
\[

$$\begin{cases} 24x + 18y \le 225 \\ x + y \ge 10 \end{cases}$$

\]
</reasoning>

<answer>
<mcq-option>(A) \(18x + 24y \le 225\)
\(x + y \ge 10\)</mcq-option>
<mcq-option>(B) \(24x + 18y \ge 225\)
\(x + y \le 10\)</mcq-option>
<mcq-option>(C) \(18x + 24y \ge 225\)
\(x + y \le 10\)</mcq-option>
<mcq-correct>(D) \(24x + 18y \le 225\)
\(x + y \ge 10\)</mcq-correct>
</answer>

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"Mathematics",
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Answer:

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<reasoning>

Define variables and quantity constraint

Using the Systems of Linear Inequalities knowledge point
\[

$$\begin{aligned} &x \ge 0, \quad y \ge 0 \\ &x + y \ge 10 \end{aligned}$$

\]

Define cost constraint

Using the Systems of Linear Inequalities knowledge point
\[

$$\begin{aligned} &\text{Cost of large frames: } 24x \\ &\text{Cost of medium frames: } 18y \\ &24x + 18y \le 225 \end{aligned}$$

\]

Combine into system of inequalities

Using the Systems of Linear Inequalities knowledge point
\[

$$\begin{cases} 24x + 18y \le 225 \\ x + y \ge 10 \end{cases}$$

\]
</reasoning>

<answer>
<mcq-option>(A) \(18x + 24y \le 225\)
\(x + y \ge 10\)</mcq-option>
<mcq-option>(B) \(24x + 18y \ge 225\)
\(x + y \le 10\)</mcq-option>
<mcq-option>(C) \(18x + 24y \ge 225\)
\(x + y \le 10\)</mcq-option>
<mcq-correct>(D) \(24x + 18y \le 225\)
\(x + y \ge 10\)</mcq-correct>
</answer>

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"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Algebra",
"Systems of Linear Inequalities"
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