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in this video we will see how to use the idea of free variables to write down the complete solution to a system of linear equations that has infinitely many solutions lets first review how row reduction helps us solve a system of linear equations the process starts by taking a system writing its corresponding Augmented matrix and then applying the row reduction algorithm since row reduction neither create nor destroy solutions we know that the system of equations corresponding to the row reduced matrix has the same solutions as the original system in this case the solutions to the original system are x equals minus 5 in y equals 7 since this is the only solution we know that this system has a unique solution lets apply the same process to the system X plus 2y equals 9 and 2x plus 4y equals 18 first we write down the Augmented matrix and then row reduce the rows of the row reduced matrix correspond to the equations X plus 2y equals 9 and 0 equals 0 this is different than the previous