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The video tutorial explains how to use free variables to find the complete solution to a system of linear equations with infinitely many solutions. By first reviewing how row reduction helps solve linear equations, the process involves creating an Augmented matrix, applying the row reduction algorithm to obtain the row reduced matrix which preserves solutions. The original system's solutions were x=-5 and y=7, showing a unique solution. Applying the same process to a new system results in equations X+2y=9 and 0=0, indicating a different outcome.
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