Question

What causes a system of two equations in two variables to have no solution? What causes a system of two equations in two variables to have infinitely many solutions?

Answer 1

You obtain the solution of a system of equations of wo equations in two variables when both equations meet. This means that both lines of each equations intersect at some point. Based on this information, you have:
 1) System of two equations in two variables with no solution:
 This is known as "Inconsistent system", where both lines are parallel, which means that never intersect each other. Therefore, there is no solution.
 2) System of two equations in two variables with infinitely many solutions:
 This is known as "Dependant system". In this type of system both equations have the same line. So, they have infinitely many solutions.