The 2-by-2 system is simple enough so that you can solve it by any means you know. However, the relations among N(A), C(A), solvable RHS, and all solutions, which should show up in the graphs, are the same for systems of any size or shape. Consider the following system of equations.
3x₁ - 2x₂ = b₁
6x₁ - 4x₂ = b₂

You will sketch two different graphs. You need relatively accurate graphs in order to answer part (f) which is the main point of this whole exercise.
(a) If we write the above system using matrix notations Ax = 5, what is the coefficient matrix A, what is 7, and what is b?
(b) What does it mean to say that the system is homogeneous? What is the definition of the nullspace of a given matrix? Describe the nullspace N(A) of this particular A in any way you can (pictures, equations, words, ...).
(c) What is the definition of the column space of a given matrix? Describe the column space C(A) of this particular A in any way you can.
(d) Choose three different right-hand vectors b ≠ 0 for any of which the system does NOT have a solution. On your first graph, sketch N(A), C(A), and these three b's.
(e) Choose two other different right-hand vectors b ≠ 0 for which the system DOES have a solution. Solve the system for each B's. How many solutions are there for each b? On your second graph, sketch N(A), C(A), these two b's, and all solutions corresponding to each b.
(f) Study your two graphs and notice any special relationship(s) among various objects. Complete the following sentences.
i. In order for At = 5 to have a solution, the right-hand vectors b (Hint: relate to C(A) and/or N(A)).
ii. In the case that Ac = b is solvable, all its solutions form C(A) (Hint: relate to C(A) and/or N(A)).