# System of linear equations-unique, infinitely many, or no solutions

by kingwinner
Tags: equationsunique, infinitely, linear, solutions
 P: 1,270 Suppose we have a system of 2 equations in 2 unknowns x,y: a1x+b1y=c1 a2x+b2y=c2 If the determinant of [a1 b1 a2 b2] is nonzero, then the solution to the system exists and is unique. [I am OK with this] If the determinant of [a1 b1 a2 b2] is zero, this does not distinguish between the cases of no solution and infinitely many solutions. To gain some insight, we need to further check the determinatnts of [a1 c1 a2 c2] and [b1 c1 b2 c2] If both determinants are zero, then the system has infinitely many soltuions. If at least one of the two determinants are nonzero, then the system has no solution. =================================== Is the bolded part correct or not? I don't see why it is true. How can we prove it? Thanks for any help!
 P: 1,270 Can someone "confirm or disprove" the bolded part, please? [my PDE book seems to be applying these ideas from linear algebra to study the solvability of initial value problems for quasilinear partial differential equations, but I can't find those results in my linear algebra textbooks other than the result "det(A) is not 0 <=> solution exists and is unique"]

 Related Discussions General Math 3 Calculus & Beyond Homework 2 Calculus & Beyond Homework 6 Linear & Abstract Algebra 14 Calculus & Beyond Homework 4