Above, I provide practice row reducing a matrix by hand. You can either choose a random matrix to reduce (choosing the dimenions and rank), you can enter a custom matrix (which may be useful when working through homework problems), or you can play the Adventure mode, where you learn new skills and scale up in difficults. The par shown is the fewest row operations needed to reach reduced row echelon form from that starting matrix, so it doubles as a rough measure of how much work the system will take.
Once you press play, you may apply one elementary operation at a time. Note, these preserve all the solutions as we are not changing the underlying system. For example, swapping rows simply just changes the order of the equations we want to simultaneously satisfy.
When you think the matrix is in reduced row echelon form, submit it. If it is, you will see the solution set and if applicable, have the ability to adjust free variables. Try to be systematic when solving. Swap to get pivots along the diagonal (when possible), and add multiples of rows to eliminate non pivots.
write-up to follow