I know there are methods out there but what do you think is the best or easiest method(s) of solving a system of equations.
Depends on the coefficients of the equation.
If in one of the equations I can solve for one of the variables without having to express it in fractions, I use substitution
3x + 7y = 9
4x + y = -3 ------> y = -4x-3, now sub that in the other equation
If not, I use elimination, that is, changing one or both of the equations to get a common coefficient for one of the variables, then subtracting or adding them to "eliminate" that variable.
2x+5y=6 ---times 3 -----> 6x+15y=18
3x-7y=1 ---times 2 -----> 6x-14y=2
now subtract them etc