7 small Decoration Christmas Balls and 2 Large Balls cost $24
4 small balls and 3 large balls cost $23 What is the cost of each item?

We can solve this problem by setting up a system of equations. Let:

- $ x $ be the cost of a small decoration Christmas ball,
- $ y $ be the cost of a large decoration Christmas ball.

From the problem, we can derive the following equations:

1. $ 7x + 2y = 24 $  (cost of 7 small and 2 large balls is $24),
2. $ 4x + 3y = 23 $  (cost of 4 small and 3 large balls is $23).

We will solve this system of equations using substitution or elimination. I'll use the elimination method:

### Step 1: Eliminate one variable
To eliminate one variable, let's multiply the first equation by 3 and the second equation by 2 to align the coefficients of $ y $.

\[
3(7x + 2y) = 3(24) \quad \Rightarrow \quad 21x + 6y = 72
\]
\[
2(4x + 3y) = 2(23) \quad \Rightarrow \quad 8x + 6y = 46
\]

### Step 2: Subtract the second equation from the first
Now subtract the second equation from the first to eliminate $ y $:

\[
(21x + 6y) - (8x + 6y) = 72 - 46
\]
\[
13x = 26
\]
\[
x = 2
\]

### Step 3: Solve for $ y $
Substitute $ x = 2 $ into one of the original equations, say $ 4x + 3y = 23 $:

\[
4(2) + 3y = 23
\]
\[
8 + 3y = 23
\]
\[
3y = 23 - 8
\]
\[
3y = 15
\]
\[
y = 5
\]

### Final Answer:
- The cost of a small ball is **$2**.
- The cost of a large ball is **$5**.

Would you like a more detailed breakdown of the steps or have any questions?

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Here are 5 related questions:
1. How would you solve this system if one equation had only one variable?
2. Can you solve this problem using the substitution method instead?
3. How would the solution change if the total cost was different but the ball counts remained the same?
4. What happens if the number of large balls was zero in both cases?
5. How would a graph of the equations look, and where would the solution lie?

**Tip:** When solving systems of equations, always double-check by plugging your solution into the original equations.

7 small Decoration Christmas Balls and 2 Large Balls cost $24. 4 small balls and 3 large balls cost $23. What is the cost of each item?

Learn how to find the cost of small and large Christmas balls by solving a system of linear equations. The problem involves key algebraic concepts like the elimination method, making it perfect for students in Grades 8-10.

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