To add to Subhotosh's comments:
Simple elimination method: Combine any two lines in such a way that the first variable coefficient is zero. For example, Subtract [1] from [2] to get the new line 2; subtract [1] from [3] to get the new line 3. For lines [4] and [5] you need to multiply first, then subtract.
Starting goal: get a 1 in the first row, left column (the 1,1 position); get all zeroes underneath that 1.
Ultimate goal (Method 1): get all 1s on the diagonal from upper left to lower right; get all zeroes under those 1s.
Method 1: At this point you have solved for one of your variables – the one in the bottom row, right hand side. You can now use this variable to solve for the others.
Method 2: Subhotosh continued working to get all zeroes above the diagonal 1s, thus yielding the complete solution that way.