Question

Write and solve a system of equations to answer the question.
Will buys 12 small balloons and 1 large balloon for a total of $30. Abigail buys 8 small balloons and 2 large balloons for a total of $28.
How much does each large balloon cost?
$

Answer 1

To solve the problem, we need to define the variables for the cost of the balloons.

Let:
- \( S \) = cost of one small balloon
- \( L \) = cost of one large balloon

Based on the information given:

1. Will's purchase of 12 small balloons and 1 large balloon for $30 can be expressed with the equation:
\[
12S + 1L = 30
\]
or simply:
\[
12S + L = 30 \quad \text{(Equation 1)}
\]

2. Abigail's purchase of 8 small balloons and 2 large balloons for $28 can be expressed with the equation:
\[
8S + 2L = 28
\]
or simplified:
\[
8S + 2L = 28 \quad \text{(Equation 2)}
\]

Now we have the following system of equations:

1. \( 12S + L = 30 \)
2. \( 8S + 2L = 28 \)

Next, we can simplify Equation 2 by dividing everything by 2:
\[
4S + L = 14 \quad \text{(Equation 3)}
\]

Now we have:
1. \( 12S + L = 30 \) (Equation 1)
2. \( 4S + L = 14 \) (Equation 3)

To eliminate \( L \), we can subtract Equation 3 from Equation 1:
\[
(12S + L) - (4S + L) = 30 - 14
\]
This simplifies to:
\[
8S = 16
\]
Solving for \( S \):
\[
S = 2
\]

Now that we have the cost of a small balloon, we can substitute \( S = 2 \) back into one of the original equations to solve for \( L \). We'll use Equation 3:
\[
4(2) + L = 14
\]
This simplifies to:
\[
8 + L = 14
\]
Subtracting 8 from both sides gives:
\[
L = 6
\]

Thus, the cost of each large balloon is:
\[
\boxed{6}
\]

To summarize, the cost of a small balloon is $2 and the cost of a large balloon is $6.