Write a system of equations to describe the situation below, solve using any method, and fill in the blanks.

Question

Russell and his friends like to collect and trade cards from a certain combat card game. Russell used his allowance to purchase 5 booster packs and 5 premade decks, which included a total of 175 cards. For his birthday, he received 2 booster packs and 8 premade decks, which included a total of 226 cards. How many cards come in every booster pack and every premade deck?

Each booster pack has
cards, and each premade deck has
cards.

Answer 1

Let's define the variables first:

• Let \( b \) be the number of cards in one booster pack.
• Let \( d \) be the number of cards in one premade deck.

We can set up the following system of equations based on the information provided:

1. For Russell's purchase: \[ 5b + 5d = 175 \]

2. For his birthday gift: \[ 2b + 8d = 226 \]

Now, we can simplify the first equation:

\[ 5b + 5d = 175 \implies b + d = 35 \quad \text{(divide the entire equation by 5)} \]

Now our system looks like:

1. \( b + d = 35 \) (Equation 1)
2. \( 2b + 8d = 226 \) (Equation 2)

Next, we can substitute \( d \) from Equation 1 into Equation 2.

From Equation 1, we can express \( d \) as: \[ d = 35 - b \]

Now, substitute this expression into Equation 2:

\[ 2b + 8(35 - b) = 226 \] Distributing the 8 gives: \[ 2b + 280 - 8b = 226 \] Combining the \( b \) terms gives: \[ -6b + 280 = 226 \] Subtracting 280 from both sides: \[ -6b = -54 \] Dividing by -6: \[ b = 9 \]

Now that we have \( b \), we can find \( d \) using Equation 1:

\[ b + d = 35 \implies 9 + d = 35 \] Subtracting 9 from both sides: \[ d = 26 \]

So, we found that each booster pack has 9 cards, and each premade deck has 26 cards.

Thus, to fill in the blanks:

Each booster pack has 9 cards, and each premade deck has 26 cards.