I was reading Linear Algebra by Hoffman and Kunze and I encountered some concepts which were arguably not explained or expounded upon thoroughly. I will present my questions below, and I am looking for answers that do not invoke concepts like determinant, vector spaces etc. Note that this is an introductory chapter that assumes no knowledge of the above concepts.

Question 1: Consider a linear system A with k equations. If we form a new equation by taking a linear combination of these k equations, then any solution of A is also a solution of this new equation. Why is this so?

Question 2a: Consider two linear systems A and B. If each equation of A is a linear combination of the equations of B, then any solution of B is also a solution of A. Why is this so?

Question 2b: If each equation of A is a linear combination of the equations of B, then it is not necessary that any solution of A is also a solution of B. Why is this so?

Question 1: Consider a linear system A with k equations. If we form a new equation by taking a linear combination of these k equations, then any solution of A is also a solution of this new equation. Why is this so?

Question 2a: Consider two linear systems A and B. If each equation of A is a linear combination of the equations of B, then any solution of B is also a solution of A. Why is this so?

Question 2b: If each equation of A is a linear combination of the equations of B, then it is not necessary that any solution of A is also a solution of B. Why is this so?