Question

writing and solving systems using elimination: practice
scarlet and paula are on a vacation. they went to a store to buy some souvenirs for their friends back home. scarlet bought and two shirts for $81 and paula bought one pair of sunglasses and five shirts for $105.
what is the cost of one pair of sunglasses and one shirt?

the cost of one pair of sunglasses is $15 and the cost of one shirt is $18.

the cost of one pair of sunglasses is $12 and the cost of one shirt is $23.

the cost of one pair of sunglasses is $18 and the cost of one shirt is $15.

the cost of one pair of sunglasses is $23 and the cost of one shirt is $12.

Explanation:

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<reasoning>

Formulate the system of equations

Using the Formulating Systems of Equations knowledge point
\[

$$\begin{cases} x + 2y = 81 \\ x + 5y = 105 \end{cases}$$

\]
where \(x\) is the cost of one pair of sunglasses and \(y\) is the cost of one shirt.

Solve the system using elimination

Using the Systems of Equations Elimination knowledge point
\[

$$\begin{aligned} (x + 5y) - (x + 2y) &= 105 - 81 \\ 3y &= 24 \\ y &= 8 \end{aligned}$$

\]

Solve for the remaining variable

Using the Systems of Equations Elimination knowledge point
\[

$$\begin{aligned} x + 2(8) &= 81 \\ x + 16 &= 81 \\ x &= 65 \end{aligned}$$

\]

Re-evaluate with corrected transcription

Using the Formulating Systems of Equations knowledge point
Let's re-read the cropped text: "Scarlet bought [one pair of sunglasses] and two shirts for $81" is actually "Scarlet bought [three pairs of sunglasses] and two shirts for $81" or similar. Let's test the given options to find the correct system:

• Option 1: \(x = 15, y = 18\).

If Scarlet bought 3 sunglasses and 2 shirts: \(3(15) + 2(18) = 45 + 36 = 81\). (Matches!)
Paula bought 1 pair of sunglasses and 5 shirts: \(1(15) + 5(18) = 15 + 90 = 105\). (Matches!)
Thus, the correct values are \(x = 15\) and \(y = 18\).

Verify the correct option

Using the Formulating Systems of Equations knowledge point
\[

$$\begin{aligned} &\text{Sunglasses: } \$15 \\ &\text{Shirt: } \$18 \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-correct>(A) The cost of one pair of sunglasses is $15 and the cost of one shirt is $18.</mcq-correct>
<mcq-option>(B) The cost of one pair of sunglasses is $12 and the cost of one shirt is $23.</mcq-option>
<mcq-option>(C) The cost of one pair of sunglasses is $18 and the cost of one shirt is $15.</mcq-option>
<mcq-option>(D) The cost of one pair of sunglasses is $23 and the cost of one shirt is $12.</mcq-option>
</answer>

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Answer:

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<reasoning>

Formulate the system of equations

Using the Formulating Systems of Equations knowledge point
\[

$$\begin{cases} x + 2y = 81 \\ x + 5y = 105 \end{cases}$$

\]
where \(x\) is the cost of one pair of sunglasses and \(y\) is the cost of one shirt.

Solve the system using elimination

Using the Systems of Equations Elimination knowledge point
\[

$$\begin{aligned} (x + 5y) - (x + 2y) &= 105 - 81 \\ 3y &= 24 \\ y &= 8 \end{aligned}$$

\]

Solve for the remaining variable

Using the Systems of Equations Elimination knowledge point
\[

$$\begin{aligned} x + 2(8) &= 81 \\ x + 16 &= 81 \\ x &= 65 \end{aligned}$$

\]

Re-evaluate with corrected transcription

Using the Formulating Systems of Equations knowledge point
Let's re-read the cropped text: "Scarlet bought [one pair of sunglasses] and two shirts for $81" is actually "Scarlet bought [three pairs of sunglasses] and two shirts for $81" or similar. Let's test the given options to find the correct system:

• Option 1: \(x = 15, y = 18\).

If Scarlet bought 3 sunglasses and 2 shirts: \(3(15) + 2(18) = 45 + 36 = 81\). (Matches!)
Paula bought 1 pair of sunglasses and 5 shirts: \(1(15) + 5(18) = 15 + 90 = 105\). (Matches!)
Thus, the correct values are \(x = 15\) and \(y = 18\).

Verify the correct option

Using the Formulating Systems of Equations knowledge point
\[

$$\begin{aligned} &\text{Sunglasses: } \$15 \\ &\text{Shirt: } \$18 \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-correct>(A) The cost of one pair of sunglasses is $15 and the cost of one shirt is $18.</mcq-correct>
<mcq-option>(B) The cost of one pair of sunglasses is $12 and the cost of one shirt is $23.</mcq-option>
<mcq-option>(C) The cost of one pair of sunglasses is $18 and the cost of one shirt is $15.</mcq-option>
<mcq-option>(D) The cost of one pair of sunglasses is $23 and the cost of one shirt is $12.</mcq-option>
</answer>

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