Visualising simultaneous linear equations

In two or 3 variables there is a clear geometrical relationship. A linear equation in 3 variables describes a plane in 3D space. Solutions to the one equation could be anywhere on that two dimensional surface. For example, the equation 2x+4y-z=3 represents a plane perpendicular to the vector (2,4,-1).

Add another equation and you are adding another plane. Unless it is parallel to the first one, the two equations now define a solution space which is a line, where the two planes intersect. Add one more plane and there is only a single point which is the solution to all 3 equations.