https://file.aiquickdraw.com/mathbot/file/1730221036662guo274a5.jpg

To solve this problem, let's set up a system of equations based on the information provided.

### Given Information
1. Lyla buys **30 balloons** in total.
2. She buys **foil balloons** at $5.49 each and **toy balloons** at $2.29 each.
3. She spends a total of **$123.10**.

### Variables
Let:
- $ f $ = the number of foil balloons.
- $ t $ = the number of toy balloons.

### Step 1: Set Up Equations
From the information given, we can write two equations:

1. The total number of balloons:
   \[
   f + t = 30
   \]

2. The total cost of the balloons:
   \[
   5.49f + 2.29t = 123.10
   \]

### Step 2: Solve the System of Equations

#### Substitute $ t = 30 - f $ (from the first equation) into the second equation:

\[
5.49f + 2.29(30 - f) = 123.10
\]

#### Expand and simplify:

\[
5.49f + 68.7 - 2.29f = 123.10
\]

\[
(5.49 - 2.29)f = 123.10 - 68.7
\]

\[
3.2f = 54.4
\]

\[
f = \frac{54.4}{3.2} = 17
\]

#### Substitute $ f = 17 $ back into the first equation to find $ t $:

\[
17 + t = 30
\]

\[
t = 30 - 17 = 13
\]

### Solution
Lyla bought:
- **17 foil balloons**
- **13 toy balloons**

Would you like further details or have any questions about this solution?

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### Related Questions
1. How would the solution change if each foil balloon cost $6.00 instead?
2. How can we verify if our solution satisfies both equations?
3. What if the total cost was $150.00 instead of $123.10?
4. How could we solve this system of equations using matrix methods?
5. How can we interpret the solution graphically?

### Tip
When setting up equations from word problems, label your variables clearly and match each part of the problem to ensure all conditions are met.

Lyla buys 30 balloons. She buys foil balloons for $5.49 each and toy balloons for $2.29 each. She pays a total of $123.10 for the balloons. Write a system of linear equations, and find the number of foil balloons f and the number of toy balloons t she buys.

Learn how to solve a system of linear equations to determine the number of foil and toy balloons purchased with specific prices and total cost. This problem involves algebraic techniques and is suitable for students in Grades 8-10.

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