Method #1: Graphing Calculator and/or Tables

Consistency:
You can easily predict the consistency of a system of equations through this method. Just graph the equations on your graphing calculator and see the intersections. If they do not intersect, they are parallel, therefore inconsistent. If they do intersect and have at least one solution, then they are consistent.

Limitations:
Some or a few old graphing and/or scientific calculators that can make tables or graphs for systems of equations may only allow up to 2 systems of equations at the same time. For old graphing calculators and some scientific calculators that can do tables, they may sometimes have a limit on how much they can display for a table. (Like for example: it can get an “insufficient memory” error if it shows more than 20 numbers on a table). Also, most graphing calculators may only be allowed to graph equations in only specific forms. (Like slope intercept form or standard form). The limitations mostly depends on the calculator you use.

Advantages:
Doesn't require much effort or any work to do since your graphing calculator pretty much does the work for you. Unlike other methods for solving systems of equations, this method can work with big decimals or fractions without problems or much difficulty. Also, you can check your answers if they are correct and you are least likely to get the wrong answer (depends if you put the right equation in the right calculator).

Disadvantages:
Depending on your graphing calculator, it may allow only 2 systems of equations to use at the same time. Also some graphing calculators require the equation to be in a specific form. (Like for example: slope intercept form (y= mx + b) or standard form (ax + by = c)) Also in most math tests that requires you to show all of your work, showing work off this method may not be enough to get full credit even if you answered the question correctly however, tests that are only multiple choice (like the SAT) may not be a problem.

Ideal type of system that this method applies to:
Depending on your graphing calculator, this mostly applies to 2 (or more if your calculator can do it) systems of equations in one specific form of equation depending on what your calculator can do.  

Example (Showing all work for this method may be impossible):
2x+ y = 1
3x+ y = 2
On my Ti-nspire CX (Not CAS), I clicked on menu, then algebra, then Solve system of linear equations, then I plugged in both equations and the result I got was (1, -1). Since this method uses a calculator, it is nearly impossible to show work since the calculator does the
work for you, which is why this is not recommended for tests that requires you to show all work for full credit.

Method #2: Elimination

Consistency:
You can predict whether the solutions are inconsistent or consistent. If you were to solve for the system of equations and end up getting a value with no variables but only numbers that’s untrue or does not make any sense (Such as 0+0=1 as the end result), then they are parallel and therefore inconsistent. If you were to end up getting a true value as a result but it ends up with no variables (Such as 0+0=0 as the end result), then they are the same line. If you are able to solve for the equation and end up with a variable that equals a number (such as x+0=2 which means x=2), then they intersect at a certain point and they have a solution, therefore, they are consistent.

Limitations:
If you were to get systems of equations but all in different forms, you can still change them all to standard form and solve for it. Even if you get crazy decimals or fractions or gigantic numbers in the equation, it will still be do-able even though it may be difficult. However, for non-terminating decimals or irrational numbers, it may be impossible to do most of them (depends on the number). If you end up having trouble with making 2 or more irrational numbers the same so you can cross them out, it may be nearly impossible to complete. Basically, about almost any systems of equations are possible, however, irrational number can be nearly impossible to solve for.

Advantages:
Some advantages of this method are the lack of limitations for this method and yet it is mostly simple multiplication and/or addition and/or subtraction. This method is very simple and yet a very useful method for solving simple or even complex equations.

Disadvantages:
Sometimes, some systems of equations that you encounter may be some crazy fractions or huge decimals (or irrational numbers) and this method may not be very easy to use in a case like that. It could also be impossible to do for irrational numbers. Also for 3 systems of equations, you may likely get an incorrect answer without noticing and things may get sloppy or difficult however, it is possible to do.

Ideal type of system that this method applies to:
Mostly a system of 2 equations with rational numbers would be the ideal type for this method. A system of 3 equations and irrational numbers can also work; however, it may be very difficult to do.

Example:
x+ 2y = 2
2x+ y = 3
Step 1: 2(x + 2y = 2) --> 2x + 4y = 4
           1(2x + y = 3) --> 2x + y = 3
Step 2: 2x + 4y = 4
             2x+ y = 3 
Subtract them to --> 0 + 3y = 1 
Divide by 3 which will give a final answer of --> y = 1/3
Step 3:
x+ 2(1/3) = 2 --> x + (2/3) = 2 --> x= 4/3

Final Answer: (4/3, 1/3)

Method #3: Substitution

Consistency:
You can predict the consistency of a system of equations through this method. Similar to elimination, if the final solution you get when you solve for one of the variables has no variables and an untrue statement (Like: 0=1), then they are parallel therefore inconsistent. If the final solution you get has no variables but a true statement (like 0=0), then they are the same line. If the final solution you get has a variable equal to a number, (like x=2) then it has a solution, therefore it is consistent.

Limitations:
Similar to Elimination, I think this method does not have any limitations however, irrational numbers (or non-terminating decimals) can possibly be impossible to do or extremely difficult.

Advantages:
Some advantages of this method are that it lacks limitations and it is do-able with all types of systems of equations. Also, this method directly substitutes for a variable then solves for it unlike elimination where you have to multiply them (depending on the equations) and then cross them out first.

Disadvantages:
If you were to encounter irrational numbers and big decimals, it will be very sloppy and hard to solve for however it is sometimes do-able but impossible to do in most cases depending on the decimal whether it is terminating at some point or not. For 3 systems of equations, it may also be a little challenging to do since there is a possible chance to mess up on a step without noticing however, it is still possible to do.

Ideal type of system that this method applies to:
This mostly applies to systems of equations with rational numbers and terminating decimals.

Example:
2x+ 3y = 3
x+ 6y = 2
Step 1: x + 6y = 2 --> x = 2 – 6y
Step 2: 2(2 – 6y) + 3y = 3 --> 4 – 12y + 3y = 3 --> 4 – 9y = 3 --> -9y = -1 --> y = 1/9
Step 3: 2x + 3(1/9) = 3 --> 2x + (3/9) = 3 --> 2x = (8/3) --> x = 4/3

Final Answer: (4/3, 1/9)

Method #4: Augmented Matrix

Consistency:
Similar to the other methods, you can easily figure out whether the solution is consistent or inconsistent by this method. If you end up getting a solution where 0 is not equal to 0, (like 0=1), then they are parallel, therefore inconsistent. If you do end up getting a solution where 0 is equal to 0, then they are the same line. If you get a solution where 1 is equal to a number in the matrix, then that means x is equal to that number which therefore means that it’s consistent. (Like x = 2).

Limitations:
Some limitations for this method is that if you were to ever encounter irrational numbers and non-terminating decimals in the systems of equations, it may be very difficult or nearly impossible to set the number in the 2nd row to be the same as the number in the 1st row. Other than that, all other rational numbers, fractions, and terminating decimals should work without a problem and should be at least possible to solve.

Advantages:
A very fast an easy method to use to solve for a system of 2 equations. This method requires only like 1 or 2 steps to complete for 2 systems of equations. If you are in a rush at a test that wants you to solve for a system of equations with any method, then this method is probably the right one to use at a situation like that since it will help save some time.

Disadvantages:
Using this method for 3 systems of equations may require some more steps to do and it may get confusing at times. Also if you encounter systems of equations with irrational numbers (like non-terminating decimals), then it could be very difficult or even impossible to do with this method.

Ideal type of system that this method applies to:
This mostly applies to 2 or 3 systems of equations with rational numbers (or at least terminating decimals).

Example:
(NOTE: Matrices may not show correctly since this post editor is not capable to show a matrix)
2x+ 4y = 5
x+ 3y = 2

Step 1: 2 4 5 --> 2R2 – R1 --> 2 4 5 --> 2y = -1 --> y = -1/2
              1 3 2                               0 2 -1 
Step 2: x + 3(-1/2) = 2 --> x = 7/2

Final Answer: (7/2 , -1/2)

Method # 5: Cramer’s rule

Consistency:
You can easily predict the consistency of a system of equations through this method. When you try to solve a system of equations and you end up getting an “undefined” answer (For example: Dividing by 0) for the values of x and y, then they are parallel (or could be the same line depending on the equations), therefore inconsistent. If you solve for the systems of equations and you do end up getting a normal value
(Like for example: x = 2), then they are consistent since they have a solution.

Limitations:
If you encounter irrational numbers, it could be extremely difficult or possibly impossible to solve for. However, everything else should be possible to complete without a problem.

Advantages:
One amazing thing about this method is the way you can quickly get a value for x and y that’s just a number out of a matrix by this method in about less than 3 steps. Just get some determinants, and simply divide them for x and y which I will show in detail at the example for this method.

Disadvantages:
Can be difficult or impossible to use this method for irrational and non-terminating decimals like most other methods.

Ideal type of system that this method applies to:
This usually applies to any 2 systems or 3 systems of equations with rational numbers (or at least terminating decimals).

Example:
x + 2y = 3
3x + y = 1

Step 1: 1 2  --> 1(1) – 2(3) = -5
              3 1
Step 2: x = 3 2  --> 3(1) – 2(1) = 1  Then: x = -1/5
                    1 1 
Step 3: y = 1 3  --> 1(1) – 3(3) = -8 Then: y = 8/5
                    3 1

Final Answer: (-1/5, 8/5)

Method #6: Inverse Matrix

Consistency:
You can easily predict the consistency of systems of equations through this method. If you were to get the inverse of a matrix and then multiply it by the original matrix and end up getting a value that is undefined or impossible to get (Or a “singular matrix” error on most calculators), then it is undefined because the lines are either parallel or the same. If you do get a normal value for x and y (Like x = 2), then it is consistent.

Limitations:
Like most other methods, irrational numbers and non-terminating decimals could be hard to deal with through this method. It may or may not be possible depending on the numbers. Other than that, it is possible to use this method with any systems of equations.

Advantages:
A pretty quick and easy method to use mostly with the 2 x 2 matrices. This method doesn’t require much things
or effort to do.

Disadvantages:
For 3 systems of equations, getting the inverse for a 3 x 3 matrix is very challenging unless you were to use a graphing calculator that is capable of doing this.

Ideal type of system that this method applies to:
This applies to 2 or 3 systems of equations with rational numbers.

Example:
x+ 3y = 2
4x+ 2y = 3

Step 1: 1 3 x  = 2
              4 2 y     3
 
Step 2: 1 3  (To the power of -1) * 1 3 x = 1 3 (To the power of -1) * 2
              4 2                                         4 2 y    4 2                                        3

Step 3: x  = 1/2  --> x = 1/2 and y = 1/2
              y     1/2

Final Answer: (1/2, 1/2)

This is about it. One problem is that I'm afraid that Matrices may not show correctly here but it should be at least understandable and decent. Thank you for reading. Thats if you actually read it. XD Any feedback will be appreciated.