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This video tutorial demonstrates how free variables can be used to find the complete solution to a system of linear equations with infinitely many solutions. By applying row reduction to the augmented matrix of the system, we can find that the original system has a unique solution of x = -5 and y = 7. For a different system, row reduction results in equations x + 2y = 9 and 0 = 0, indicating that this system has infinitely many solutions.