The Nature of Solutions — practice questions

Solving a system produces the equation $0 = 0$ at the final step. State how many solutions the system has, what this means geometrically, and what must be given as the answer.

Determine the nature of the solutions of $2x - y + z = 4$, $4x - 2y + 2z = 8$, $x + y - z = 1$, giving the full solution set and a geometric description.

Prove that a system of linear equations cannot have exactly two solutions, and use the argument to explain why the solution set of any consistent linear system must be a point, a line, or a plane.

Two equations of a system are $3x - 6y + 9z = 12$ and $x - 2y + 3z = 7$. What does this tell you about the system's solutions, and why?

For what value of $p$ does the system $x + y + 2z = 3$, $2x - y + z = 1$, $3x + 0y + 3z = p$ have infinitely many solutions? What happens for other values of $p$, and what is the geometry in each case?

Analyse the system $x + y + z = 1$, $2x + 3y + kz = 3$, $x + ky + 3z = 2$ completely: determine all values of $k$ giving a unique solution, infinitely many, or none, with the general solution or the inconsistency in each special case.