To find the Thévenin equivalent resistance $R_{th}$ of the circuit to the left of the resistor $R_L$, we follow these steps:

1. Remove $R_L$: Since we’re calculating the Thévenin equivalent resistance, we’ll remove $R_L$ from the circuit, which leaves only the resistors and the 3V source.

2. Deactivate the Voltage Source: We replace the 3V source with a short circuit (since an ideal voltage source is replaced by a short when calculating $R_{th}$).

3. Calculate Equivalent Resistance:

   • After shorting the 3V source, the two 20-ohm resistors are in series, and the two 30-ohm resistors are also in series. This forms two branches in parallel with each other and the 50-ohm resistor in the middle.
   • We’ll start by finding the equivalent resistance for each branch and then combine them.

   Step-by-Step Calculation:

   • Left Branch: $20 \Omega + 30 \Omega = 50 \Omega$
   • Right Branch: $20 \Omega + 30 \Omega = 50 \Omega$

   Now, the left branch (50 ohms), the middle 50-ohm resistor, and the right branch (50 ohms) are all in parallel. The equivalent resistance of three parallel 50-ohm resistors is calculated as:

   $\frac{1}{R_{th}} = \frac{1}{50} + \frac{1}{50} + \frac{1}{50} = \frac{3}{50}$

   Solving for $R_{th}$:

   $R_{th} = \frac{50}{3} \approx 16.67 \, \Omega$

Thus, the Thévenin equivalent resistance $R_{th}$ is approximately 16.67 ohms.

Would you like further details on this calculation, or do you have additional questions?

Here are some related questions for practice:

1. How would you find the Thévenin equivalent voltage of the same circuit?
2. If the voltage source were 5V instead of 3V, how would the Thévenin resistance change?
3. How would adding another resistor in parallel with the existing 50-ohm resistor affect $R_{th}$?
4. How do you find the equivalent resistance when resistors are arranged in a combination of series and parallel?
5. Why is it necessary to deactivate the source when calculating $R_{th}$?

Tip: Remember, for Thévenin resistance calculations, deactivate all sources: replace voltage sources with short circuits and current sources with open circuits.