On Monday, a computer manufacturing company sent out three shipments. The first order, which contained a bill for $116,000, was for 4 Model II, 6 Model VI, and 10 Model IX computers. The second shipment, which contained a bill for $73,000, was for 8 Model II, 3 Model VI, and 5 Model IX computers. The third shipment, which contained a bill for $82,000, was for 2 Model II, 9 Model VI, and 5 Model IX computers. What does the manufacturer charge for each Model VI computer?

To find out the cost of each type of computer, we can set up a system of equations based on the information provided.

Let's denote:
- $ x $ as the price of one Model II computer,
- $ y $ as the price of one Model VI computer, and
- $ z $ as the price of one Model IX computer.

From the problem, we have the following information:

1. The first order: $ 4x + 6y + 10z = 116,000 $
2. The second order: $ 8x + 3y + 5z = 73,000 $
3. The third order: $ 2x + 9y + 5z = 82,000 $

We now have a system of three equations:

\[
\begin{aligned}
4x + 6y + 10z &= 116,000 \quad \text{(1)} \\
8x + 3y + 5z &= 73,000 \quad \text{(2)} \\
2x + 9y + 5z &= 82,000 \quad \text{(3)}
\end{aligned}
\]

### Step 1: Solve the System of Equations

We can use the method of elimination or substitution to solve this system. I'll proceed with the elimination method.

#### Multiply equation (1) by 2 to simplify later elimination:
\[
8x + 12y + 20z = 232,000 \quad \text{(4)}
\]

#### Subtract equation (2) from equation (4):
\[
(8x + 12y + 20z) - (8x + 3y + 5z) = 232,000 - 73,000
\]
\[
9y + 15z = 159,000 \quad \text{(5)}
\]

#### Multiply equation (3) by 2:
\[
4x + 18y + 10z = 164,000 \quad \text{(6)}
\]

#### Subtract equation (1) from equation (6):
\[
(4x + 18y + 10z) - (4x + 6y + 10z) = 164,000 - 116,000
\]
\[
12y = 48,000
\]

\[
y = 4,000
\]

### Step 2: Find the Cost of Model VI
We have found that the manufacturer charges **$4,000** for each Model VI computer.

### Step 3: Verification
We can substitute $ y = 4,000 $ back into the original equations to find $ x $ and $ z $, but since the question only asked for the price of the Model VI computer, we have the answer.

Would you like me to continue with finding the prices of the other models or any other details?

---

Here are 5 related questions that can expand your understanding:

1. How would you solve for the price of Model II and Model IX computers using this information?
2. What method would be best if there were more than three variables or equations?
3. Can this method be used to solve real-world business scenarios beyond just computer shipments?
4. How can you verify the solution to ensure there are no calculation errors?
5. What happens if one of the equations was incorrect? How would you identify and correct it?

**Tip:** When solving a system of equations, always check your solution by substituting the values back into the original equations to confirm they satisfy all conditions.

On Monday, a computer manufacturing company sent out three shipments. The first order, which contained a bill for $116,000, was for 4 Model II, 6 Model VI, and 10 Model IX computers. The second shipment, which contained a bill for $73,000, was for 8 Model II, 3 Model VI, and 5 Model IX computers. The third shipment, which contained a bill for $82,000, was for 2 Model II, 9 Model VI, and 5 Model IX computers. What does the manufacturer charge for each Model VI computer?

Learn how to solve a system of linear equations to find the price of each computer model in a real-world shipment scenario. This problem involves algebraic concepts like systems of equations and elimination methods, suitable for students in Grades 10-12. In this example, we determine the price of Model VI computers using three real-life shipment data points.

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