So from memory, and without guarantee that terminology corresponds to English mathematics

• Every hyperplane can be expressed as a linear equation derived from the normal vector
• hence multiple hyperplanes form a System of linear equations
• solutions to those equations are the intersections of the corresponding hyperplanes
• if you have less equations than variables you get a solution set in the form of a linear equation with free variables.
• the later is just another hyperplane /line of dimension of mentioned free variables, (a point is just a dimension 0 solution, i.e no free variables)

• there are various numeric algorithms to solve equation systems represented as a matrix, like Gauss elimination. (Others might be better in terms of limiting rounding errors)
• they also produce a solution set for under defined systems (less rows than columns)

There are some generalizations applied when dealing with linear optimization methods:

• the equations are just the boundaries of a < "less than" inequalities, i.e. like all points "left" all boundaries are valid solutions,
• the intersections of those boundary equations are the corners of a convex polygon/polytope then, whose body represents solutions
• a further target/weight function might be given to choose the "optimal" solution in this solution set
• finally: sometimes only integer solutions are sought after, while the exact solution is non-integer. (You don't wanna buy a third of an olive)

I hope this helps you finding the words to express your questions

Again I'm rusted, and might have confused some (English) terminology.

thank you LanX for the teminology explainer.

Don't expect those to be "rational", it's long known that people tend to drive many kilometers because they are baited by a 1€ discount on some beer or such.

Marketing is more about psychology.