- #1
Krushnaraj Pandya
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Homework Statement
Consider three equations in three variables, To check their consistency usually the adjoint of the matrix (from coefficients of x,y and z are found) and the inverse is found for the solution. I have heard of cramer's rule but it has a similar complexity.
These same kind of questions are repeated every year in the JEE which makes it very important for students like me to figure out something fast since there's no time to be calculating inverses and adjoints and risk mistakes as well.
As a sample- Here's a question from 2017 JEE Mains
Find the set of distinct values of b for which the following equations has no solution.
x+y+z=1; x+ay+z=1; ax+by+z=0
Homework Equations
Any pertaining to algebra
The Attempt at a Solution
I thought of some ways which might be faster but I don't know if they are foolproof-
1)Solving them pairwise- (easier for a solution- but I don't know how consistency can be checked)
2) Using 3d geometry, these are equations of planes. I'm not sure how exactly we can proceed using that though
3) As for the sample question, setting determinant=0 gives a=1 which means two of the equations are identical- x+y+z can't be 0 and 1 at the same time (I suppose but I'm not sure) so b is 1. Is that the right way to go about it?