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rows to eliminate variables until the system is in triangular form, which can then be easily solved by back-substitution. For example, consider the following system of equations: 3x + 2y - z = 1 2x - 2y + 4z = -2 -x + 0.5y - z = 0 The augmented matrix for this system is: [ 3 2 -1 | 1 ] [ 2 -2 4 |