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Let's stick letters in for the numbers:

ax = b ... ( 1 / a ) times ax = ( 1 / a ) times b ... x = b / a

Now, let's rewrite this guy so he's not a fraction:

ax = b ... a^( -1 ) times ax = a^( -1 ) times b ... x = a^( -1 ) times b
This notation is completely division free.

Why did we go through all this?

Because we are dealing with matrices and not numbers!  And we can't do division with matrices.