I believe I understand both topics individually: When asked if a linear system spans a certain R^n, the question is, "can any point be reached in that dimensional plane?" Linear combination is multiplying a vector with a scalar and adding that to another vector being multiplied by a scalar and creating a linear system to solve to see if theres a unique solution or set of solutions.
My question is, how is it that finding a set of solutions or a unique solution, through the use of linear combination, will tell you if the system spans that dimensional plane? I'm having a hard time understanding how know that theres a set of numbers that solved all 3 equations means that it spans the plane. If anyone can explain this part in layman's terms, that would be great.