2. Water is flowing through a network of pipes (in thousands of cubic meters per hour), as show in the figure. Set up but do not solve the system of equations for this network. Indicate which junction corresponds to which equation in your work. Then rewrite all equations in a system of equations where variables on one side and constants on the other. 500 600 X3 X4 X5 500 600 X6 X7

Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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**Problem Statement:**

Water is flowing through a network of pipes (in thousands of cubic meters per hour) as shown in the figure. Set up, but do not solve, the system of equations for this network. Indicate which junction corresponds to which equation in your work. Then rewrite all equations in a system of equations where variables are on one side and constants on the other.

**Diagram Explanation:**

The diagram consists of three junctions connected by pipes with flow rates represented by variables \(x_1\) through \(x_8\). The flow rates are accompanied by known inflow and outflow values at certain junctions:

- **Junction 1:** 
  - Inflow: \(600\)
  - Outflows: \(x_1\) (to Junction 2 next to it) and \(x_3\) (downwards to Junction 3).

- **Junction 2:**
  - Inflow: \(x_1\) (from Junction 1 above it)
  - Outflows: \(x_2\) (to the right) and \(x_4\) (downwards to Junction 4).

- **Junction 3:**
  - Inflow: \(x_3\) (from Junction 1 above it)
  - Outflows: \(x_5\) (to Junction 4 next to it) and \(x_7\) (downwards, with an outflow of 500).

- **Junction 4:**
  - Inflow: \(x_4\) (from Junction 2 above it), \(x_5\) (from Junction 3 above it)
  - Outflow: \(x_6\) (downwards, with an outflow of 500).

**Flow Requirements:**

At each junction, the total inflow must equal the total outflow. This principle results in the following equations:

1. For Junction 1:
   - \(600 = x_1 + x_3\)

2. For Junction 2:
   - \(x_1 = x_2 + x_4\)

3. For Junction 3:
   - \(x_3 = x_5 + x_7\)
   - \(x_7 = 500\)

4. For Junction 4:
   - \(x_4 + x_5 = x_6 + 500\
Transcribed Image Text:**Problem Statement:** Water is flowing through a network of pipes (in thousands of cubic meters per hour) as shown in the figure. Set up, but do not solve, the system of equations for this network. Indicate which junction corresponds to which equation in your work. Then rewrite all equations in a system of equations where variables are on one side and constants on the other. **Diagram Explanation:** The diagram consists of three junctions connected by pipes with flow rates represented by variables \(x_1\) through \(x_8\). The flow rates are accompanied by known inflow and outflow values at certain junctions: - **Junction 1:** - Inflow: \(600\) - Outflows: \(x_1\) (to Junction 2 next to it) and \(x_3\) (downwards to Junction 3). - **Junction 2:** - Inflow: \(x_1\) (from Junction 1 above it) - Outflows: \(x_2\) (to the right) and \(x_4\) (downwards to Junction 4). - **Junction 3:** - Inflow: \(x_3\) (from Junction 1 above it) - Outflows: \(x_5\) (to Junction 4 next to it) and \(x_7\) (downwards, with an outflow of 500). - **Junction 4:** - Inflow: \(x_4\) (from Junction 2 above it), \(x_5\) (from Junction 3 above it) - Outflow: \(x_6\) (downwards, with an outflow of 500). **Flow Requirements:** At each junction, the total inflow must equal the total outflow. This principle results in the following equations: 1. For Junction 1: - \(600 = x_1 + x_3\) 2. For Junction 2: - \(x_1 = x_2 + x_4\) 3. For Junction 3: - \(x_3 = x_5 + x_7\) - \(x_7 = 500\) 4. For Junction 4: - \(x_4 + x_5 = x_6 + 500\
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