1. **Problem Statement:** We have a control system with forward path transfer function $G(s) = \frac{s+1}{s(s+2)}$ and feedback path transfer function $H(s) = \frac{s+3}{s+4}$. The system has a negative feedback loop. 2. **Goal:** Push $H(s)$ to the right past the summing junction and find the equivalent system. 3. **Key Concept:** When moving a block past a summing junction, the transfer functions must be adjusted to maintain the same overall system behavior. 4. **Original system:** The output $Y(s)$ is given by: $$Y(s) = G(s) \cdot \left(R(s) - H(s)Y(s)\right)$$ 5. **Rearranging:** $$Y(s) + G(s)H(s)Y(s) = G(s)R(s)$$ $$Y(s)(1 + G(s)H(s)) = G(s)R(s)$$ $$\Rightarrow \frac{Y(s)}{R(s)} = \frac{G(s)}{1 + G(s)H(s)}$$ 6. **Moving $H(s)$ past the summing junction:** To push $H(s)$ to the right, we redefine the system so that the summing junction subtracts $Y(s)$ multiplied by a new function $H_{new}(s)$ after $G(s)$. 7. **Equivalent system:** The new feedback function after $G(s)$ is: $$H_{new}(s) = G(s)H(s)$$ 8. **Calculate $H_{new}(s)$:** $$H_{new}(s) = \frac{s+1}{s(s+2)} \times \frac{s+3}{s+4} = \frac{(s+1)(s+3)}{s(s+2)(s+4)}$$ 9. **Final equivalent system:** - Forward path: $G(s)$ remains the same. - Feedback path after $G(s)$: $H_{new}(s) = \frac{(s+1)(s+3)}{s(s+2)(s+4)}$. This means the summing junction now subtracts $H_{new}(s)Y(s)$ after $G(s)$. **Answer:** The equivalent system has the same $G(s)$ block, and the feedback block moved to the right with transfer function: $$H_{new}(s) = \frac{(s+1)(s+3)}{s(s+2)(s+4)}$$