https://file.aiquickdraw.com/mathbot/file/1726789880661gf45yqpj.jpg

Let's break down the problem shown in the diagram and work on setting up the linear system.

The goal is to determine the unknown flow rates (\(x_1, x_2, x_3, x_4, x_5, x_6\)) for each pipe segment in the network based on the inflow and outflow constraints at each node (A, B, C, D, and E). The problem asks us to create a matrix equation of the form:

\[
A \vec{x} = \vec{y}
\]

where:

- \(\vec{x}\) is the vector of unknowns \([x_1, x_2, x_3, x_4, x_5, x_6]^T\),
- \(\vec{y}\) is the vector of known inflows and outflows at each node \([25, 175, 200, 150, 200]^T\).

### Step 1: Write Flow Balance Equations at Each Node

For each node, the sum of incoming flows should equal the sum of outgoing flows:

- **Node A**:
  - Incoming: 25 (from outside)
  - Outgoing: \(x_1\) to E, \(x_2\) to B
  - Flow balance: \( 25 = x_1 + x_2 \)

- **Node B**:
  - Incoming: \(x_2\) from A, \(x_4\) from D
  - Outgoing: \(x_6\) to C
  - Flow balance: \( x_2 + x_4 = x_6 + 175 \)

- **Node C**:
  - Incoming: \(x_6\) from B
  - Outgoing: 200 (to outside)
  - Flow balance: \( x_6 = 200 \)

- **Node D**:
  - Incoming: \(x_3\) from E
  - Outgoing: \(x_4\) to B, \(x_5\) to C
  - Flow balance: \( x_3 = x_4 + x_5 + 150 \)

- **Node E**:
  - Incoming: 200 (from outside)
  - Outgoing: \(x_1\) to A, \(x_3\) to D
  - Flow balance: \( 200 = x_1 + x_3 \)

### Step 2: Set Up the System in Matrix Form

Now, we can represent the system of equations from the flow balances in matrix form.

\[
A \vec{x} = \vec{y}
\]

- The matrix \(A\) will contain the coefficients of \(x_1, x_2, ..., x_6\).
- The vector \(\vec{x}\) will be \([x_1, x_2, x_3, x_4, x_5, x_6]^T\).
- The vector \(\vec{y}\) will be \([25, 175, 200, 150, 200]^T\).

Would you like me to proceed with forming the full matrix equation for \(A\) and solving it? Let me know if you need details or have any questions.

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Here are 5 related questions for further exploration:

1. What methods can be used to solve a system of linear equations like this one?
2. How does the structure of the matrix \(A\) reflect the flow relationships in the network?
3. What is the significance of conservation of mass in this type of flow network?
4. How do inflow and outflow conditions affect the uniqueness of the solution?
5. What computational tools can solve systems of linear equations efficiently?

**Tip:** When analyzing flow networks, always ensure that incoming and outgoing flows at each node are balanced to maintain mass conservation principles.

Set up a linear system whose solution provides the unknown flow rates for the given network of pipes in the oil refinery diagram.

Set up a linear system to find unknown flow rates in a network of pipes at an oil refinery. This problem involves key linear algebra concepts like matrix equations, flow balances, and conservation of mass. Suitable for students in Grades 10-12 and college-level courses.

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