I don't prefer to go by most textbook definitions for matters like these.
Instead, the first thing I try to recongize is the conditions of the reaction. Is the reaction under polar acidic or polar basic conditions?
Then I try and recognize what sort of tranformation is occuring. In this case, you're positive it's a substitution or elimination reaction.
Those are always my first two steps.
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(1) If it's polar basic, chances are it's either an SN2- or E2-like mechanism. To figure out which one it is, I look at the structure and the attacking atom.
Is the structure (substrate) hindered? Most of the time, if it has three groups attached it's E2.
Is the attacking atom a good nucleophile? Is it a good base? Weak nucleophile? Weak base? If it's a good nucleophile but a poor base, it's most certainly SN2. (Look at the chart I've provided as a .doc file)
(2) If it's polar acidic, chances are it's either SN1 or E1.
Of course in this case it's not a matter of substrate hindrance but substrate stabilization.
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Solvent effects, as movies said, has a large effect on kinetics. There are many factors that influence what mechanism the reaction will undergo. Substrate dependence, nucleophilicity vs. basicity, kinetics, as well as thermodynamics all affect substitution and elimination mechanisms.
If you're looking at two starting materials, one is obviously your alkyl halide and the other is a potential nucleophile/base, ask yourself the same questions I did. Solvent effects stabilize transition states and/or starting materials, but are not necessarily the most important thing in determining which mechanism will prevail.
Let me end with an example. Let's say we have a reaction between t-BuBr and MeOH. These are polar acidic conditions, and the substrate is fairly hindered. We can rule out SN2. We can also rule out E2 because MeOH is not a strong base (or even remotely basic for that matter).
So we've narrowed it down to either SN1 or E1. This is a 3° substrate, so a resulting alkene (after E1) would be thermodynamicallly favorable.
In this case solvent effects are of very little importance because it just effects kinetics. Because the possible mechanisms are E1 or SN1, the stabilization effects are the same.
We can conclude that the overal mechanism will be SN1, because the poor basicity of MeOH overrules the thermodynamic favorability of the resulting alkene after E1. This does not mean E1 will not occur. But given the circumstances, the amounts of alkene obtained will probably be really small.
As you can see, I've cirumvented the issue of E2 vs. SN1 based on the argument of basicity of MeOH, or lack thereof.
Remember, solvent effects influence TSs and energies of activation more than anything else. Don't get too bogged down by these effects, as they're not always the most important thing in determining how a reaction proceeds.