Question

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

An employee at a party store is assembling balloon bouquets. For a graduation party, he assembled 6 small balloon bouquets and 3 large balloon bouquets, which used a total of 126 balloons. Then, for a Father's Day celebration, he used 246 balloons to assemble 7 small balloon bouquets and 8 large balloon bouquets. How many balloons are in each bouquet?

Answer 1

Let \( x \) represent the number of balloons in a small bouquet, and \( y \) represent the number of balloons in a large bouquet. We can set up a system of equations based on the information provided:

From the graduation party, where 6 small and 3 large bouquets use a total of 126 balloons, we have the equation:

\[
6x + 3y = 126
\]

From the Father's Day celebration, where 7 small and 8 large bouquets use a total of 246 balloons, we have the second equation:

\[
7x + 8y = 246
\]

Now we can simplify the first equation. Divide all terms in the first equation by 3:

\[
2x + y = 42 \quad \text{(Equation 1)}
\]

Now we write the second equation as it is:

\[
7x + 8y = 246 \quad \text{(Equation 2)}
\]

Next, we can use the elimination method to solve this system of equations. First, we will express \( y \) from Equation 1:

From Equation 1:

\[
y = 42 - 2x \quad \text{(Substituting for y)}
\]

Now substitute \( y \) in Equation 2:

\[
7x + 8(42 - 2x) = 246
\]

Distribute \( 8 \):

\[
7x + 336 - 16x = 246
\]

Combine like terms:

\[
-9x + 336 = 246
\]

Now, subtract 336 from both sides:

\[
-9x = 246 - 336
\]
\[
-9x = -90
\]

Dividing both sides by -9 gives:

\[
x = 10
\]

Now substitute \( x = 10 \) back into the expression for \( y \):

\[
y = 42 - 2(10)
\]
\[
y = 42 - 20
\]
\[
y = 22
\]

Thus, the number of balloons in each bouquet is:
- Small bouquet: **10 balloons**
- Large bouquet: **22 balloons**

To summarize:

\[
\boxed{10} \text{ for small bouquets and } \boxed{22} \text{ for large bouquets.}
\]