Question

Who's Right and Why?

While sitting in class you overheard one of your classmates say, "This equation has infinite solutions, because when I put
it in Desmos it gave me parallel lines so I know it makes a conditional equation." Realizing that the student is incorrect,
what would you say to help your classmate get their facts straight?

Answer 1

If the two equations result in parallel lines when graph there are no solutions to the system of equations since there will never be intersections of the two equations.

Explanation:

For this problem, it's important to note that the classmate has mistaken no solutions to a system of equations with infinitely many solutions to the system.

For the system to have infinitely many solutions, then the equations within the system would need to be identical, meaning that at any point on the graph, the equations always have "crossings" since they are the exact same.

For the system to have no solutions, then the equations within the system would never be able to have "crossings". The most common occurrence is when you have a set of parallel lines, as described in this problem. Two parallel lines will never "cross" by definition.

For the system to have a limited number of solutions, then the equations within the graph would only "cross" a finite amount of times within a graph.

Cheers.