1. **State the problem:** We have a pipe network with nodes E, D, A, B, C and flows $x_1$ through $x_6$ between them. Inputs are 200 at E, 150 at D, 25 at A, and output is 200 at C. We need to find the possible flows by writing and solving flow conservation equations. 2. **Write flow conservation equations:** - At node E: Input 200 enters, flows out are $x_1$ (to A) and $x_3$ (to D). Conservation means input equals output: $$200 = x_1 + x_3$$ - At node D: Input 150 plus flow $x_3$ from E enter; flows out are $x_4$ (to B) and $x_5$ (to C): $$150 + x_3 = x_4 + x_5$$ - At node A: Input 25 plus flow $x_1$ from E enter; flow out is $x_2$ (to B): $$25 + x_1 = x_2$$ - At node B: Inputs are $x_2$ (from A) and $x_4$ (from D); output is 175 leaving B: $$x_2 + x_4 = 175$$ - At node C: Inputs are $x_5$ (from D) and $x_6$ (from B); output is 200 leaving C: $$x_5 + x_6 = 200$$ - Also, flow $x_6$ flows from B to C, so $x_6$ is an unknown. 3. **Express variables and solve step-by-step:** From node A: $$x_2 = 25 + x_1$$ From node B: $$x_2 + x_4 = 175 \implies (25 + x_1) + x_4 = 175 \implies x_4 = 175 - 25 - x_1 = 150 - x_1$$ From node D: $$150 + x_3 = x_4 + x_5 \implies 150 + x_3 = (150 - x_1) + x_5 \implies x_3 + x_1 = x_5$$ From node E: $$200 = x_1 + x_3 \implies x_3 = 200 - x_1$$ Substitute $x_3$ into previous: $$200 - x_1 + x_1 = x_5 \implies x_5 = 200$$ From node C: $$x_5 + x_6 = 200 \implies 200 + x_6 = 200 \implies x_6 = 0$$ 4. **Summary of flows:** - $x_5 = 200$ - $x_6 = 0$ - $x_3 = 200 - x_1$ - $x_4 = 150 - x_1$ - $x_2 = 25 + x_1$ 5. **Constraints:** All flows must be non-negative: - $x_1 \geq 0$ - $x_3 = 200 - x_1 \geq 0 \implies x_1 \leq 200$ - $x_4 = 150 - x_1 \geq 0 \implies x_1 \leq 150$ So $x_1$ can vary between 0 and 150. **Final answer:** $$\boxed{\begin{cases} x_1 = t, \quad 0 \leq t \leq 150 \\ x_2 = 25 + t \\ x_3 = 200 - t \\ x_4 = 150 - t \\ x_5 = 200 \\ x_6 = 0 \end{cases}}$$