Op amp design that gives \$V_{out} = -2 *V_{1} + 3*V_{2} - 3 V_{3}\$
Despite Dave's warning that figuring out the resistor values for a single opamp implementation will be non-trivial, we'll take a crack at it.  The trick is to evolve the circuit in small steps, so that each step is pretty simple.  Start with the basic inverting amplifier using a single opamp:



This is real basic, with the gain being -R1/R2.  If you're not sure where that came from, look up "inverting amplifier" and I'm sure there will be lots written about this out there.  Due to the feedback around the amplifier, the circuit will keep pin 2 at whatever voltage pin 3 is, which in this case is 0.  Assuming a ideal opamp, the node at pin 2 has 0 impedance to 0 volts.  The current flowing onto that node thru R2 is proportional to the voltage at IN.  That same current must also flow thru R1 (ideal opamp, infinite input impedance), but it flows thru R1 left to right, so the right end is lower.  The net effect is inversion.

Just to pick something, let's say R1 is 100 k&Omega;.  If R2 is also 100 k&Omega; then the gain is -1, if it is 50 k&Omega; then the gain is -2 (same current makes twice the voltage accross R1 as it does accross R2).

No consider that current from various places can be dumped onto the pin 2 node independently since that node is always at 0 V.  Each of these shows up accross R1, with the total R1 voltage resulting from the sum of these currents.  You could make a second input, for example, that had a different negative gain:



Since R1 is 2x larger than R2, the voltage at V1 will be amplified by 2x (and inverted).  Similarly, since R3 is 3x R1, the voltage at V2 will be amplified by 3x.  So far we have

&nbsp;&nbsp;Vout = -2*V1 - 3*V2

Now think of the common non-inverting opamp amplifier:



Here the gain is a little different.  R1 and R2 form a voltage divider for the feedback signal, and the gain ends up being the attenuation factor of the divider.  The gain in this case is:

&nbsp;&nbsp;G = (R1 + R2) / R2

Now go back to the 2-input inverting amp above and see that we actually have such a arrangement from the positive input's point of view.  V1 and V2 are voltage sources and therefore have 0 impedance.  Impedance-wise R2 and R3 are in parallel and are the bottom resistor of the voltage divider for the feedback.  In other words, we already have a non-inverting amplifier, just that so far we've grounded its input.  R2 in the positive amp gain equation is actually the parallel combination of R2 and R3 of the 2-input inverting amp.  33.3 k&Omega; // 50 k&Omega; = 20 k&Omega;.  Plugging that into the positive gain equation above, we get a gain of 6 from pin 3 to Vout.

But the problem states you want a gain of 3, not 6.  No problem, just attenuate by 2 before presenting it to the opamp, which will multiply it by 6.  Overall, you get 3 just like you want.  A final circuit to realize all three gains is:



R4 and R5 attenuate V3 by 2, then the circuit multiplies that by 6, so overall the gain from V3 to Vout is 3.

Since all these effects are linear, we can consider each one individually knowing they will all just add up.  Put another way, analyze what happens so the signal from each input while holding the other inputs at 0.

I just realized I have V2 and V3 flipped from your question, but what I wrote above is self-consistant and should answer your question well enough.