See this Wikipedia article, especially Method 2. It uses matrix algebra to solve a system of linear equations, but this system can be solved using any method, such as the elimination of variables, which I believe is taught in school. You could put d = -1 (or any other non-zero number).
Find a Cartesian equation for the plane P in R3 that passes through the points A(1,0,0), B(0,3,0) and C(0,0,4).
I there any way to do this question without using the cross product?
Form vectors and . Now you have one point (A with its radius vector ) and two vectors that lie in the same plane. Using them you can form a parametric equation of that plane, describing a radius vector of an arbitrary point in that plane.
Componentwise it looks like this.
And from there it is easy to form the equation of the plane, just substituting and in the first equation with the expression from second and third eq.