The equivalent resistance between the points A and B in the following circuit is $$\frac{x}{5}$$Ω. The value ofx is________.

step 1: assume currents

Let total current from source = i

At node A, current splits:
top branch =$$i_1$$
bottom branch = $$i-i_1$$

step 2: apply loop equation (top loop)

$$6i_1+3(2i_1-i)=3(i-i_1)$$

simplify:

$$6i_1+6i_1-3i=3i-3i_1$$

$$12i_1-3i=3i-3i_1$$

step 3: apply KVL for left path

$$3(i-i_1)+6i_1=1$$

substitute $$i_1$$:

$$3\left(i-\frac{2}{5}i\right)+6\left(\frac{2}{5}i\right)=1$$

$$3\left(\frac{3}{5}i\right)+\frac{12}{5}i=1$$

$$\frac{9}{5}i+\frac{12}{5}i=1$$

step 4: equivalent resistance

$$R_{eq}=Vi=\frac{1}{\frac{5}{21}}=\frac{21}{5}Ω$$

given form: