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The problem provided involves solving a system of equations based on the cost and composition of three different nut mixes, A, B, and C. Let's break it down step by step.

### Information Given:
1. The table provides the weight (in hundreds of grams) of cashews, macadamias, and Brazil nuts required for each mix:
   - **Mix A**: 5 units of cashews, 2 units of macadamias, 3 units of Brazil nuts.
   - **Mix B**: 4 units of cashews, 4 units of macadamias, 1 unit of Brazil nuts.
   - **Mix C**: 6 units of cashews, 1 unit of macadamias, 3 units of Brazil nuts.
   
2. The costs per kilogram of each mix are:
   - 1 kg of **Mix A** costs $12.50,
   - 1 kg of **Mix B** costs $12.40,
   - 1 kg of **Mix C** costs $11.70.

3. We are tasked with:
   - Determining the cost per kilogram for **each type of nut** (cashews, macadamias, Brazil nuts).
   - Finding the cost per kilogram of a custom mix containing **400 grams of cashews, 200 grams of macadamias, and 400 grams of Brazil nuts**.

### Step 1: Define Variables
Let:
- $ x $ be the cost per kilogram of **cashews**.
- $ y $ be the cost per kilogram of **macadamias**.
- $ z $ be the cost per kilogram of **Brazil nuts**.

### Step 2: Formulate System of Equations
Based on the weight compositions and the cost for each mix, we can form the following system of equations:

\[
5x + 2y + 3z = 12.50 \tag{1} \quad \text{(Mix A)}
\]
\[
4x + 4y + 1z = 12.40 \tag{2} \quad \text{(Mix B)}
\]
\[
6x + 1y + 3z = 11.70 \tag{3} \quad \text{(Mix C)}
\]

### Step 3: Solve the System of Equations
Now we will solve this system of equations to find the values of $ x $, $ y $, and $ z $.

Let me calculate this.The cost per kilogram of each type of nut is as follows:
- **Cashews** ($x$): \$0.98 per kg
- **Macadamias** ($y$): \$1.78 per kg
- **Brazil nuts** ($z$): \$1.34 per kg

### Step 4: Calculate the Cost of the Custom Mix
The custom mix contains:
- 400 grams (or 0.4 kg) of cashews,
- 200 grams (or 0.2 kg) of macadamias,
- 400 grams (or 0.4 kg) of Brazil nuts.

The total cost of the custom mix is given by:

\[
\text{Total cost} = 0.4x + 0.2y + 0.4z
\]

Let me calculate this next.# Custom mix calculation: 0.4 kg cashews, 0.2 kg macadamias, 0.4 kg Brazil nuts
custom_mix_cost = 0.4 * solution[x] + 0.2 * solution[y] + 0.4 * solution[z]
custom_mix_costThe cost of the custom mix containing 400 grams of cashews, 200 grams of macadamias, and 400 grams of Brazil nuts is approximately **$1.29**.

Would you like more details on any of the steps? Here are some related questions to deepen your understanding:

1. How can you verify the solution of the system of equations?
2. What happens if the cost of one type of nut changes? How does it affect the overall price of the mixes?
3. Can you solve the same system of equations using matrix methods like Gaussian elimination?
4. What would the cost be for a different custom mix composition?
5. How would you modify the problem if additional nut types were introduced?

**Tip:** Understanding systems of equations is essential in many areas of mathematics, especially in economics and optimization problems!

A mixed nut company uses cashews, macadamias, and Brazil nuts to make three gourmet mixes. The table shows the weight (in hundreds of grams) of each kind of nut required to make a kilogram of each mix. Given the cost of 1 kg of Mix A, B, and C, determine the cost per kilogram of each nut and find the cost for a custom mix.

Learn how to calculate the cost per kilogram of cashews, macadamias, and Brazil nuts by solving a system of linear equations. Understand how to use the costs of different mixes and apply these values to determine the cost of a custom mix containing specific amounts of each nut. Suitable for students in Grades 10-12.

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