A customer at a store paid $64 for 3 large candles and 4 small candles. At the same store, a second customer paid $4 more than the first customer for 1 large candle and 8 small candles. The price of each large candle is the same, and the price of each small candle is the same.

How much did each candle cost?

Let's break down the problem systematically by setting up two variables:

- Let $ x $ represent the price of one large candle.
- Let $ y $ represent the price of one small candle.

### Step 1: Write the first equation based on the first customer's purchase.
The first customer bought 3 large candles and 4 small candles for $64, so we have the equation:

\[
3x + 4y = 64
\]

### Step 2: Write the second equation based on the second customer's purchase.
The second customer bought 1 large candle and 8 small candles and paid $4 more than the first customer, i.e., $64 + 4 = $68. This gives the equation:

\[
x + 8y = 68
\]

### Step 3: Solve the system of equations.

#### Equation 1: $ 3x + 4y = 64 $
#### Equation 2: $ x + 8y = 68 $

We can solve this system by substitution or elimination. I'll use substitution.

From Equation 2, solve for $ x $:

\[
x = 68 - 8y
\]

Substitute this expression for $ x $ into Equation 1:

\[
3(68 - 8y) + 4y = 64
\]

Expand and simplify:

\[
204 - 24y + 4y = 64
\]

\[
204 - 20y = 64
\]

Now, subtract 204 from both sides:

\[
-20y = 64 - 204
\]

\[
-20y = -140
\]

Solve for $ y $:

\[
y = \frac{-140}{-20} = 7
\]

So, the price of a small candle is $ y = 7 $ dollars.

### Step 4: Substitute $ y = 7 $ into Equation 2 to find $ x $.

Substitute $ y = 7 $ into $ x + 8y = 68 $:

\[
x + 8(7) = 68
\]

\[
x + 56 = 68
\]

Subtract 56 from both sides:

\[
x = 12
\]

Thus, the price of a large candle is $ x = 12 $ dollars.

### Final Answer:
- The price of a large candle is **$12**.
- The price of a small candle is **$7**.

Would you like a detailed explanation of any particular step? Here are some additional questions you might consider:

1. What method could be used if you wanted to solve this system by elimination instead of substitution?
2. How would the system change if the second customer bought a different number of candles?
3. Could this problem be solved graphically? How?
4. How do systems of linear equations apply to real-world scenarios like this one?
5. What would happen if the prices of both candles were the same?

**Tip:** Always double-check your solution by substituting the values back into the original equations!

A customer at a store paid $64 for 3 large candles and 4 small candles. At the same store, a second customer paid $4 more than the first customer for 1 large candle and 8 small candles. The price of each large candle is the same, and the price of each small candle is the same. How much did each candle cost?

Discover how to solve a system of linear equations to find the price of large and small candles. This algebra problem involves the substitution method and systems of equations, suitable for students in Grades 8-10.

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