GATE 2016 EE SET -2 Question 27

$$\frac{V_{A}-V_{1}}{1}+\frac{V_{A}-V_{2}}{4}=0$$

$$\frac{4V_{A}-4V_{1}+V_{A}-V_{2}}{4}=0$$

$$\Rightarrow \frac{5V_{A}-5V_{1}}{4}=0$$

$$\Rightarrow 5V_{A}-4V_{1}-V_{2}=0$$

$$5V_{A}=4V_{1}+V_{2}$$        $$\Rightarrow V_{A}=\frac{4}{5}V_{1}+\frac{1}{5}V_{2}$$ _______ (1)

$$\frac{V_{A}-V_{3}}{1}+\frac{V_{A}-V_{out}}{9}=0$$ $$\Rightarrow 9V_{A}-9V_{3}+V_{A}-V_{out}=0$$

$$10V_{A}-9V_{3}=V_{out}$$ ________ (2)

Substituting equation (1) into equation (2)

$$10(\frac{4}{5}V_{1}+\frac{1}{5}V_{2})-9V_{3}=V_{out}$$

$$8V_{1}+2V_{2}-9V_{3}=V_{out}$$