To do this, you must first make a 3x3 matrix with each of its columns equal to each of the vectors u, v, and w. Then, to get the desired vector^^^, you reduce the matrix you just created to the 3x3 identity matrix*** with the vector x, so my solution is this:
columns = u v w | x
row a = | -1 3 -3 | 9 | --> a*-1 --> | 1 -3 3 | -9 | --> a+3*b --> on
row b = | 0 -6 5 | -17 | --> b/-6 --> | 0 1 -5/6 | 17/6 | ---> as is ---> next
row c = | -5 -3 1 | -12 | --> c-a*5 --> | 0 -18 16 | -57 | --> c+18*b --> lines
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row a = | 1 0 1/2 | -1/2 | ----> a-c/2 ----> | 1 0 0*** | 5/2^^^ |
row b = | 0 1 -5/6 | 17/6 | --> b+5*c/6 --> | 0 1 0*** | -13/6^^^ |
row c = | 0 0 1 | -6 | ----> as is ----> | 0 0 1*** | -6^^^ |
