1. **Problem statement:**
We have a traffic network with nodes A, B, C, D and variables $x_1, x_2, x_3, x_4, x_5$ representing vehicle flows on certain roads. Given known flows and directions, we need to set up a system of equations based on flow conservation at each node and solve for $x_1, x_2, x_3, x_4, x_5$.
2. **Key principle:**
At each node, the total inflow equals the total outflow (conservation of vehicles).
3. **Set up equations at each node:**
- Node A:
Inflows: 200 (from left), 600 (from bottom)
Outflows: $x_1$ (to right), $x_2$ (to top)
Equation: $$200 + 600 = x_1 + x_2 \implies 800 = x_1 + x_2$$
- Node B:
Inflows: 400 (from left), $x_2$ (from bottom)
Outflows: 300 (down), $x_4$ (to right)
Equation: $$400 + x_2 = 300 + x_4 \implies x_4 = 400 + x_2 - 300 = 100 + x_2$$
- Node C:
Inflows: 450 (from left), $x_4$ (from top), $x_5$ (from right)
Outflows: 150 (down)
Equation: $$450 + x_4 + x_5 = 150 \implies x_4 + x_5 = 150 - 450 = -300$$
- Node D:
Inflows: $x_1$ (from left), $x_3$ (from right)
Outflows: $x_5$ (to top)
Equation: $$x_1 + x_3 = x_5$$
4. **Summarize equations:**
$$\begin{cases}
800 = x_1 + x_2 \\
x_4 = 100 + x_2 \\
x_4 + x_5 = -300 \\
x_1 + x_3 = x_5
\end{cases}$$
5. **Substitute $x_4$ from second into third:**
$$100 + x_2 + x_5 = -300 \implies x_5 = -400 - x_2$$
6. **Substitute $x_5$ into fourth:**
$$x_1 + x_3 = -400 - x_2$$
7. **From first equation:**
$$x_1 = 800 - x_2$$
8. **Substitute $x_1$ into above:**
$$(800 - x_2) + x_3 = -400 - x_2 \implies x_3 = -400 - x_2 - 800 + x_2 = -1200$$
9. **Now we have $x_3 = -1200$ (negative flow means opposite direction).**
10. **Calculate $x_5$:**
$$x_5 = -400 - x_2$$
11. **Calculate $x_4$:**
$$x_4 = 100 + x_2$$
12. **Summary of variables in terms of $x_2$:**
$$x_1 = 800 - x_2, \quad x_3 = -1200, \quad x_4 = 100 + x_2, \quad x_5 = -400 - x_2$$
13. **Interpretation:**
Since $x_3$ and $x_5$ are negative for any $x_2$, the assumed directions for these flows are opposite to the actual flow.
14. **If we want all flows positive, choose $x_2$ to satisfy constraints or reorient directions accordingly. For example, if $x_2 = 0$:
$$x_1 = 800, x_3 = -1200, x_4 = 100, x_5 = -400$$
**Final answer:**
$$\boxed{x_1 = 800 - x_2, \quad x_2 = x_2, \quad x_3 = -1200, \quad x_4 = 100 + x_2, \quad x_5 = -400 - x_2}$$
where $x_2$ is a free parameter representing flow on the left vertical segment between A and B.
Traffic Flows
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