Question 2(Multiple Choice Worth 1 points) \( \begin{array}{l}(05.03 \mathrm{MC}) \\ \text { The total price of a shirt and a cap is } \$ 12 \text { JIf the price of the shirt was doubled and the price of the cap was three times its original price, the total price of a shirt and a cap would be } \$ 26 \text {. } \\ \text { The following system of equations models this scenario: } \\ x+y=12 \\ 2 x+3 y=26 \\ \text { What is the price of each shirt and cap? }\end{array} \)

To solve the equations \( x + y = 12 \) and \( 2x + 3y = 26 \), where \( x \) represents the price of the shirt and \( y \) represents the price of the cap, we can use substitution or elimination methods. By rearranging the first equation, we get \( y = 12 - x \). Substituting this into the second equation gives us \( 2x + 3(12 - x) = 26 \). Simplifying, we find \( 2x + 36 - 3x = 26 \), leading to \( -x + 36 = 26 \), which gives \( x = 10 \) for the shirt. Plugging \( x \) back into the first equation finds \( y = 2 \) for the cap. The final prices are: Shirt = $10, Cap = $2. Now, let’s dig into some interesting insights! Did you know that the concept of systems of equations has been a crucial part of mathematics since ancient civilizations? The Babylonians were solving linear equations over 3,000 years ago, making them the original equation whizzes! And if you want to explore more about these equations, many textbooks dive deep into linear algebra. A fun recommendation is "Linear Algebra Done Right" by Sheldon Axler, where matrix magic can make solving equations even more engaging! It's perfect for bridging the gap to advanced topics.