very simple
suppose your equations have max 2 variables
and some free numbers
you move your free numbers to one side and the variables to the other
example 5x+2y+3 =0
4x+5y+1=0
you take the indices of every variable and write them in a line
like this:
5 2 | 3
3 5 | 1
then you add and subtract lines to make it so it looks like this
1 0
0 1
for the given example it will be
L1 = L1 - L2 (L stands for line)
so you do 5-4 , 2-5, 3-1 and get
1 -3| 2
4 5 |1
now need to make sure the 4 in the 2nd line becomes a 0
so L2 = L2 - 4L1 (4 times line 1 subtracted from line 2)
4-4 , 5-(-3)*4 , 2-(1)*4 and get
1 -3| 2
0 17| -2
now you need to make the 17 to a 1
so L2 = 1/17 L2
1 - 3|2
0 1|-2/17
now you need the -3 in L1 to become a 0
so L1 = L1 +3L2
1+0*3 ,-3+1*3, 2+(-2/17)*3
1 0 | 1 11/17 (one and eleven seventeenths)
0 1|-2/17
so the answer to the above given equation set would be
x= 1 11/17
y = -2/17
if the number of equations equals to the number of variables and there are no duplicate lines (lines sum of, subtraction of will give a 0 0 0 ... |0 line)
then there is only one answer
if there are more variables than lines, there is an infinite number of answers
and usually a scheme of what type of elements fit in the group of answers
the same principle is applied to equations with more than 2 variables, and more than 2 lines
Good Luck!
