1. Reduced Row Echelon Form
My personal favorite!

2. Elementary Row Operations
We can perform three elementary row operations on matrices:
Multiplying a row by a constant.
Switching two rows.
Adding a constant times a row to another row.
To solve a system of equations, first put into augmented matrix form.
This system
4x – 5y + 3z = 2
x – y – 2y = -6
4x – 4y – 14z = 18
BECOMES

3. To row reduce a matrix:
Perform elementary row operations to yield a "1" in the first row, first column.
Create zeros in all the rows of the first column except the first row by adding the first row times a constant to each other row.
Perform elementary row operations to yield a "1" in the second row, second column.
Create zeros in all the rows of the second column except the second row by adding the second row times a constant to each other row.
Perform elementary row operations to yield a "1" in the third row, third column.
Create zeros in all the rows of the third column except the third row by adding the third row times a constant to each other row.
Continue this process until the first m×m entries form the identity matrix.

5. TA DAH!
Notice the identity matrix in the first 3 rows and columns.
Now we are ready to read the results!
Putting this matrix back into equation form:
1x + 0y + 0z = -123, x = -123
0x + 1y + 0z = y = -103
0x + 0y + 1z = z =
(-123, -103, -7) is the ordered triple solution.

6. Reduced Row Echelon Form (RREF) is on your calculator too
Reduced Row Echelon Form (RREF) is on your calculator too. Let’s try another problem, this time doing things by calculator:
x – 2y + 3z = 9
-x + 3y = -4
2x – 5y + 5z = 17
Put into calculator as 3 x 4 matrix.
Go to main screen, go to matrix menu, choose math, RREF (matrix name)
I love it!

7. This is method gives you the most info, even if the system comes out with no solution or an infinite number of solutions:

8. This method is so wonderful that someone has already built an App that makes use of it. If you have PLY SMLT2 you are ready to do a bunch easily. If not, you will have to create an appropriately sized augmented matrix, perform RREF on the matrix, and life is still good!