I am grading linear algebra homework, and a not-insubstantial percentage of students attacked the problem of determining whether or not [2,6,6] was in the span of [-1,2,3], [3,4,2], and [-9,-2,5] by converting this into the system of 3 equations in three unknowns:

{ –x + 3 y – 9 z = 2,   2 x + 4 y – 2 z = 6,   3 x + 2 y + 5 z = 6 }

…and then, rather than row-reduce a matrix as we have done for the LAST EIGHT WEEKS, proceeded to solve this system by isolating the x in the first equation and substituting that into the second, and then isolating the y in that equation and substituting it into the third equation, and…

Clearly, I have failed them.

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