Point of intersection of pair of straight lines

Suppose the two lines given to us are $L(x,y)=0$ and $J(x,y)=0$, where $L(x,y)=px+qy+r$ and $J(x,y)=sx+ty+z$. Then the equation of the pairs of these lines is $$LJ=0. tag{1}$$ Let us consider the line given by $$L+lambda J=0. tag{2}$$ For different values of $lambda$, we will get a line that passes through the intersection of the lines $L$ and $J$. The only point $(x,y)$ for which (2) will have a solution for every $lambda$ is when $(x,y)$ is the intersection of the lines $L$ and $J$.

Coming back to equation (1), if we take the partial derivative with regards to $x$ and $y$, then we get begin{align*} L_x, J+J_x , L & =0\ L_y, J+J_y , L & =0. end{align*} Note that $L_x,L_y,J_x,J_y$ are all constants (real numbers). Suppose say $J_x, J_y neq 0$, then we can divide by $J_x$ and $J_y$ to get begin{align*} L +lambda_1 J & =0\ L +lambda_2 J & =0. end{align*} Which is basically in the form of equation (2). So a common solution for this system will give us the solution for equation (2) that can work for all values of $lambda$, and hence the intersection point.