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How do you figure out how to determine the best method to solve the system of equations then sole the system?

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There are three methods that are usually used to solve the a system of equations.  These are graphing, substitution, and elimination.  All three methods will get the same answer, but each one has advantages and disadvantages.

Graphing:  Graphing is the best method to use when introducing a new student to solving systems of two equations in two variables, because it gives them a visiual to recognize what they are looking for.  Graphing is less exact and often takes more time than the other methods.  I only recommend graphing to find a solution if the problem comes with a graph already drawn and the intersection appears to be on an exact coordinate.

Substitution:  Substition gives that advantage of having an equation alrady written for the second variable when you find the first one.  Substitution is best used when one (or both) of the equations is already solved for one of the variables.  It also works well if one of the variables has a coefficient of 1.

Elimination:  Elimination is the method that I use almost every time.  If you are not sure which method to use, I recommend that you use elimination.  Elimination is best used when both equations are in standard form (Ax + By = C).  Elimination is also the best method to use if all of the variables have a coefficient other than 1.

I hope this gives you some idea about which method you should use.  All three methods will give the same solution.  Also, you should always check your answer in both equations to make sure it is correct.