Mathematics · essential
Consistency criterion for two variables for JEE
Test of consistency and solution of simultaneous linear equations in two or three variables using matrices
Formula
For \(a_1x+b_1y+c_1=0\) and \(a_2x+b_2y+c_2=0\): unique solution if \(\dfrac{a_1}{a_2}\ne\dfrac{b_1}{b_2}\); no solution if \(\dfrac{a_1}{a_2}=\dfrac{b_1}{b_2}\ne\dfrac{c_1}{c_2}\); infinitely many solutions if \(\dfrac{a_1}{a_2}=\dfrac{b_1}{b_2}=\dfrac{c_1}{c_2}\).
Variables: \(a_i,b_i,c_i\): coefficients and constants.
Conditions: Denominators considered only when defined; equivalent determinant form may also be used.
Testing consistency of pair of linear equations.