Simulatneous Equations And Geometrical Interpretation Flashcards
What happens if there is no UNIQUE solution to a simultaneous equation, what’s special about the determinant
It means det M = 0!
There are 3 situations for 2 x 2 simultaneous, how to get to them and identity each
Find det, if det is non 0 then it is non singular and there is A UNIQUE SOLUTION, they intersect at a poimt
If det is 0, singular = NO UNIQUE SOLUTIONS
2) if they are the SAME LINE = they on top of each other = INFINITE SOLUTIONS
3) if they are PARALLEL = never intersect = NO SOLUTIONS
How does parallel work for 2 x 2
Remember parallel is if same grsdient , and this will be true if coefficient of y and x are the same (y too because you end up dividing by y)
The fact that c part is diff means it’s just shifted a bit in y direction, thus PARALLEL
So parallel if y and x same, but c diff
If you have 3 disticnt planes what are the possibilities
2 are parallel , 1 intersects = no solutions
3 are parallel = no solutions
All form a triangular prism = no solutions
All intersect at a line, forming a sheaf = infinite soltuiosn
Or they all intersect = one solution
So if all 3 are distinct what to do
Find det M, if 0 then
- check for parallels by making coeffeicmt the same
- if can’t then it’s either a prism or a sheaf
To make a sheaf, check if lines are consistent, sub 1 with 2 and 1 with 3 and eliminate one variable, if the same then you got your infinite line of soltuojd
If not consistent then must be a triangular prism = NO SOLUTIONS
How to go about each geometrical interprwtyaion question
First check if parallel, and thus will check if same
If not check det
I’d det not 0, find inverse and solve
If 0, then check if consistent , sheaf, if not triangular prism
How to check you’re not consistent calculation is correct for triangular prims?
Because the coeffeicmt does of 2 variables will still be the same, it’s the “d” that will be different!
