Solve 3.3 = 0.765 * (1 + R1/10) for R1.

Step-by-step solution

1. 1

   Recognise the structure before touching the algebra. This is the standard output equation of an adjustable regulator or a non-inverting divider: a fixed reference voltage $0.765$ multiplied by a gain $1 + R_1/R_2$, with $R_2 = 10$. Solving for $R_1$ means undoing that gain, so the operations come off in reverse order: divide, subtract, multiply.

2. 2

   Divide both sides by the reference.

   $\frac{3.3}{0.765} = 1 + \frac{R_1}{10}$

   Keeping it exact, $\tfrac{3.3}{0.765} = \tfrac{33/10}{153/200} = \tfrac{6600}{1530} = \tfrac{220}{51} \approx 4.313725$.

3. 3

   Subtract 1 to isolate the ratio.

   $\frac{R_1}{10} = \frac{220}{51} - 1 = \frac{169}{51} \approx 3.313725$

4. 4

   Multiply by the lower resistor value.

   $R_1 = 10 \cdot \frac{169}{51} = \frac{1690}{51} \approx 33.137$

   In the usual units this is about $33.14\ \text{k}\Omega$ when $R_2 = 10\ \text{k}\Omega$ — conveniently close to the E24 standard value $33\ \text{k}\Omega$.

5. 5

   Check by substituting back. $0.765\left(1 + \tfrac{33.137255}{10}\right) = 0.765 \times 4.3137255 = 3.300000$. Using the nearest standard $33\ \text{k}\Omega$ instead gives $0.765 \times 4.3 = 3.2895\ \text{V}$, about $0.3\%$ low.