There's a law, an awesome transformation, when your using variability… you can solve all sorts of problems… easily… Reduced Row Echelon Form… Reduced Row Echelon Form… Reduced Row Echelon Form… Reduced Row Echelon Form… Solves all sorts of problems… Reduced Row Echelon Form…
Anyway, Let's say you have a bunch of variables and a bunch of equations and you want to solve for all of them. What a pain!
i.e. (not sure what this will reduce to..)
2x+4y-3z+5w+2q=4
4x-2y-8z+8w+7q=5
2x+4y-3z+6w+8q=9
6x-6y-6z+1w+2q=1
2x+4y-4z+9w+4q=5
Note, you must have n number of equations and n number of unknowns (in this case 5 equations and 5 unknowns)
Say you have 4 equations and 5 unknowns.. then you are under defined. Say you have 5 equations and 4 unknowns.. then you are over defined. So you want and equal number of equations and unknowns.
Anyway, to solve this problem throw the coefficients into a mattrix and put them into Reduced Row Echelon Form! This will solve it for any number of unknowns as long as you have enough equations to solve for them. The best part is that it involves simple addition and multiplication!