Question

why a system of equations must have at least as many equations as there are variables to have a single solution?

Answer 1

the rank of the column space= the rank of the row space, so there will be an infinite number of solutions.

Answer 2

or you can think of it like two planes cannot intersect at a point, they can only intersect in a line.

Answer 3

sorry but i dont get it can you explain more please

Answer 4

In 2 dimensions, if you have ax+by=c and px+qy=r, then these two lines will intersect once (that is, you know the point of intersection). If you had 1 line, you wouldn't know the point of intersection as there is no other line to compare it to. The point of intersection could be on any point on the line given.

Answer 5

In 3 dimensions, if you have 2 equations it is not enough. That is, ax+by+cz=d and px+qy+rz=s will not give you one point of intersection, but a line (similar to 2 dimensions + 1 equation: when you are 1 equation too few, you get a line of intersection, not a point). If you have 2 equations too few, you get a plane of intersections (e.g. in 3D having just ax+by+cz=d).

Answer 6

thanks a lot mannn you really helped me, not just helped me you saved my life lol this question had a big bones :D i appreciate it