Intersection of two lines in vector form

The most elegant formulation (IMO) uses Cramer's Rule. You can reformulate your two equations into $$\begin{bmatrix} u_1 & -u_2\\ v_1 & -v_2 \end{bmatrix} \cdot \begin{bmatrix} a \\ b\end{bmatrix} = \begin{bmatrix} x_2-x_1 \\ y_2 -y_1 \end{bmatrix}$$ and use determinants to solve for $a$.

Another nice way to think about it is areas: For segments AB and CD, the fraction along AB at which the intersection occurs is equal to the area of the triangle ACD divided by the area of the quadrilateral ACBD.

You could use the elimination method to solve the simultaneous equations rather than the substitution method. If the two lines aren't parallel then there should be no cases when you could accidentally divide by $0$.