1. The problem statement, all variables and given/known data Problem: A system of linear equations can't have exactly two solutions. Why? (a) If (x, y, z) and (X, Y, Z) are two solutions, what is another solution? (b) If 25 planes meet at two points, where else do they meet? Solution: (a) Another solution is 1/2 * (x + X, y + Y, z + Z). (b) If 25 planes meet at two points, they meet along the whole line through those two points. 2. Relevant equations N/A 3. The attempt at a solution Am I correct in thinking that a system of linear equations can't have exactly two solutions because a line in three-dimensional space must intersect at zero points or one point or a full line's worth of points and a plane in three-dimensional space must intersect at zero points, a line's worth of points or a plane's worth of points when attempting to answer the question asked before part (a)? Assuming I'm correct, I know that, for example, when I say “a line's worth” it means an infinite amount of points and that that is the same thing as “a plane's worth” etc but, I'm just expressing myself like that to clarify what I am saying. For part (a), I was thinking (x + X, y + Y, z + Z) would be another solution. Am I right? Are all answers with scalars multiplying each component and then operators summing, dividing, multiplying or subtracting those respective (transformed) components solutions as well? For part (b), is it correct to add that not only could they meet at an entire (infinite) line's worth of points but that another case is that if the planes overlap, they can meet at an entire (infnite) plane's worth of points? If something is unclear, tell me and, I will attempt to clarify it. Any input would be greatly appreciated!