(1)One of the problems with using graphing to try to solve a system of equations is trying to get an exact answer.
For example the system:
has a solution: `(142/63,115/63)` . This would be difficult to decipher from a hand-drawn graph. Even with technology, you will frequently get an approximation for the x and y values. This is useful in an engineering context, but not so much if you need to use the result in later computations.
(2) If the system is not linear (with rational coefficients), you may end up with irrational points which would be virtually impossible to describe without approximations.
(3) If the system is larger than a 2x2 (two equations in two unknowns) the dimension of the coordinate system grows. A 3x3 system is in 3-dimensional space. Finding an intersection when you are reduced to drawing (imperfectly) a 3-d representation in two dimensions is difficult. And if the system is larger, you cannot graph at all as we can not easily describe higher dimensions geometrically.