Question

A system of linear equations can't have exactly two solutions. why?

a) The system is inconsistent
b) There is infinite solution
c) It violates the triangle inequality
d) The determinant is zero

Answer 1

A system of linear equations can't have exactly two solutions due to the properties of lines in a plane, as they can only intersect at one point, be parallel with no points of intersection, or coincide with an infinite number of solutions.

Step-by-step explanation:

A system of linear equations can't have exactly two solutions because of the properties of linear systems in a two-dimensional space. This is mainly described by the potential configurations of two lines, which are the graphical representations of linear equations in two variables. There are three possibilities:

• If the lines intersect at a single point, there is exactly one solution, which represents the point of intersection.
• If the lines are parallel and distinct, there are no solutions; this condition means that the system is inconsistent.
• If the lines coincide, meaning they are the same line, there are infinitely many solutions because every point on the line satisfies both equations.

Given these scenarios, the correct answer is that a system of linear equations can't have exactly two solutions because it would violate the fundamental principles of linear equations and their graphical representation as lines in a plane.