I'm still confused about what you want, and I have trouble linking to the right pages. Likely a problem of terminology...

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)

Matrix calculations:

• 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.