Gauss-Jordan elimination solves a system of equations by pure routine. You write the coefficients and the right-hand sides into one table, the augmented matrix, and then repeat just three moves: swap two rows, divide a row by a number, and subtract a multiple of one row from another. The animation above performs exactly those moves on the numbers you type in.
The target is the identity matrix in the left half: the first row then says only x, the second only y, and the answer is sitting on the right. The route there never changes. Make the coefficient of x equal to 1 and clear x out of the other row, then do the same for y. No insight required, only order.
A swap is needed when the place that should become a 1 holds a 0. You cannot divide by 0, so you bring up the row below it. When the left half refuses to become the identity, the solution is not unique. A contradictory 0 = 3 means there is no solution; an equation collapsing to 0 = 0 means there are infinitely many. Both cases are waiting in the examples.