Q.Design an op-amp circuit to realize the output, $V_O = 3V_1-2V_2+V_3$, Assume $R_F = 10\text{ k}\Omega$.

Use an op-amp inverting adder; because it inverts, apply $-V_1$, $V_2$ and $-V_3$ at its inputs, then choose each input resistor so that $R_f/R_n$ equals the required coefficient. …

[!TLDR]

Realise the sum with a single inverting adder fed by $-V_1$, $V_2$, $-V_3$; the resistors work out to $R_1=3.33\text{ k}\Omega$, $R_2=5\text{ k}\Omega$, $R_3=10\text{ k}\Omega$.

An op-amp inverting adder gives

The required output $V_O=3V_1-2V_2+V_3$ has a $+$ sign on $V_1$ and $V_3$ and a $-$ sign on $V_2$. Since the adder inverts, we apply $-V_1$, $V_2$ and $-V_3$ to the three inputs; the leading minus sign then restores the wanted signs:

$V_O=-R_f\!\left(\dfrac{-V_1}{R_1}+\dfrac{V_2}{R_2}+\dfrac{-V_3}{R_3}\right)=\dfrac{R_f}{R_1}V_1-\dfrac{R_f}{R_2}V_2+\dfrac{R_f}{R_3}V_3$ …