If (a_1b_2-a_2b_1\neq0), what is the correct statement about the pair?

The pair has a unique solution

For equations written as \(a_1x+b_1y+c_1=0\) and \(a_2x+b_2y+c_2=0\), the expression \(a_1b_2-a_2b_1\) is the determinant of the coefficient matrix. If it is non-zero, the coefficients are not proportional. Consequently, the two lines have different slopes and cannot be parallel or identical; they must intersect at one point.

The given condition says this determinant is not zero. Therefore the system is solvable in exactly one way: there is one ordered pair \((x,y)\) satisfying both equations. This is called a unique solution, so choice C is correct. A zero determinant would require further comparison of the constant terms, because it could describe either infinitely many solutions or no solution. A non-zero determinant avoids both of those cases.

This is a Class 10 Mathematics question. Chapter: Pair of Linear Equations in Two Variables. Topic: Conditions for solvability.