Write a system of equations to describe the situation below, solve using an augmented matrix, and fill in the blanks. Pedro and his friends like to collect and trade cards from a c... Write a system of equations to describe the situation below, solve using an augmented matrix, and fill in the blanks. Pedro and his friends like to collect and trade cards from a certain combat card game. Pedro used his allowance to purchase 9 booster packs and 6 premade decks, which included a total of 360 cards. For his birthday, he received 1 premade deck, which included a total of 48 cards. How many cards come in every booster pack and every premade deck?
Understand the Problem
The question is asking to write a system of equations based on Pedro's card purchases, then solve it using an augmented matrix. It provides specific quantities related to booster packs and premade decks, which need to be represented mathematically.
Answer
The number of cards in each booster pack is $8$, and in each premade deck is $48$.
Answer for screen readers
The number of cards in each booster pack is $8$, and the number of cards in each premade deck is $48$.
Steps to Solve
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Define Variables Let ( x ) = number of cards in each booster pack
Let ( y ) = number of cards in each premade deck -
Set Up the Equations From the information given:
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For Pedro's purchases: The equation for the total number of cards from 9 booster packs and 6 premade decks is: $$ 9x + 6y = 360 $$
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For the premade deck received for his birthday: The equation for the total number of cards in the premade deck is: $$ y = 48 $$
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Form the Augmented Matrix Combine the equations into an augmented matrix form: $$ \begin{bmatrix} 9 & 6 & | & 360 \ 0 & 1 & | & 48 \end{bmatrix} $$
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Use Row Operations Perform row operations to solve the augmented matrix. Start with the second row:
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Keep it as is: $$ \begin{bmatrix} 9 & 6 & | & 360 \ 0 & 1 & | & 48 \end{bmatrix} $$
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Next, eliminate ( y ) from the first equation. Multiply the second row by 6 and subtract from the first row: $$ \begin{bmatrix} 9 & 0 & | & 360 - (6 \cdot 48) \ 0 & 1 & | & 48 \end{bmatrix} $$
- Calculate Values Calculate the result from the first row: $$ 360 - 288 = 72 $$
So, we now have: $$ \begin{bmatrix} 9 & 0 & | & 72 \ 0 & 1 & | & 48 \end{bmatrix} $$
- Solve for Variables From the first row: $$ 9x = 72 $$ gives $$ x = \frac{72}{9} = 8 $$
From the second row: $$ y = 48 $$
- Final Values The solution is: $$ x = 8, y = 48 $$
The number of cards in each booster pack is $8$, and the number of cards in each premade deck is $48$.
More Information
Pedro's card collection problem illustrates how to formulate a system of equations based on real-world scenarios and solve them using an augmented matrix. The mathematical concepts applied here can be useful in various areas such as finance and logistics.
Tips
- Not clearly defining the variables can lead to confusion, so it’s important to establish what each letter represents.
- Mixing up the coefficients in the equations or the totals can cause incorrect results; always double-check the calculations.
- Forgetting to reduce the augmented matrix fully can result in inaccurate solutions.
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