A merchant mixed 8 lb of a cinnamon tea with 2 lb of spice tea. The 10-pound mixture cost $31. A second mixture included 12 lb of the cinnamon tea and 12 lb of the spice tea. The 24-pound mixture cost $78. Find the cost per pound of the cinnamon tea and of the spice tea.

We can solve this problem using a system of linear equations. Let:

- $ x $ be the cost per pound of the cinnamon tea (in dollars),
- $ y $ be the cost per pound of the spice tea (in dollars).

### Step 1: Set up the system of equations

We are given two mixtures and their costs. From this information, we can create two equations.

- **First mixture**: 8 lb of cinnamon tea and 2 lb of spice tea, total cost $31:
  \[
  8x + 2y = 31
  \]
- **Second mixture**: 12 lb of cinnamon tea and 12 lb of spice tea, total cost $78:
  \[
  12x + 12y = 78
  \]

### Step 2: Simplify the second equation

We can simplify the second equation by dividing everything by 12:
\[
x + y = 6.5
\]

### Step 3: Solve the system of equations

Now, we have the following system:
1. $ 8x + 2y = 31 $
2. $ x + y = 6.5 $

#### Solve equation 2 for $ y $:
\[
y = 6.5 - x
\]

#### Substitute $ y = 6.5 - x $ into equation 1:
\[
8x + 2(6.5 - x) = 31
\]
Simplify the equation:
\[
8x + 13 - 2x = 31
\]
\[
6x + 13 = 31
\]
\[
6x = 31 - 13
\]
\[
6x = 18
\]
\[
x = 3
\]

#### Substitute $ x = 3 $ into $ y = 6.5 - x $:
\[
y = 6.5 - 3 = 3.5
\]

### Step 4: Conclusion

The cost per pound of the cinnamon tea is **$3**, and the cost per pound of the spice tea is **$3.50**.

Would you like further details or have any questions?

Here are five related questions for practice:
1. If the cinnamon tea cost $4 per pound, what would be the total cost of a 10-pound mixture with the same ratios?
2. How would the cost of the mixtures change if the spice tea cost $5 per pound?
3. Can you solve a similar problem with different weights of tea?
4. How would the solution change if we added a third type of tea to the mixture?
5. What happens to the system of equations if the cost of the mixtures remains the same but the ratios of tea types are different?

**Tip**: When dealing with systems of equations, always simplify first before solving—it can save time and reduce errors!

Learn how to find the cost per pound of cinnamon tea and spice tea using a system of linear equations. This example covers solving two equations: 8x + 2y = 31 and 12x + 12y = 78, using substitution. Perfect for Grades 8-10 students studying algebra and linear systems.

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