The fallacy here is thinking that having the same number of equations as variables guarantees that you will have a unique solution. However, it's possible to have no solution or to have infinitely many solutions. Here is an example of each for only 2 equations and 2 variables:
1) $x+y=1,x+y=2$ has no solutions.
2) $x+y=1,x+y=1$ has infinitely many solutions.
Having at least as many equations as variables is a necessary condition for a unique solution, but it is not sufficient. In particular, if we can take each of the equations, multiply them each by some number (not all 0), and add them all up to get a constant equation $a=b$, then there won't be a unique solution. If $a$ and $b$ are actually equal (ex: $3=3$), then the solution exists but is not unique. If $a$ and $b$ are distinct (ex: $2=9$), then there will be no solution. If there is no way to get such an equation by this process, then there will be a unique solution.
This process is called a linear combination, and the equations are called linearly dependent when some linear combination creates a constant equation, and linearly independent when no linear combination does.