Ohio Assessments for Educators (OAE) Mathematics Practice Exam

How many solutions exist when two lines intersect at a point on a graph?

1. No solutions

2. Exactly 1 solution

3. Infinitely many solutions

4. Two solutions

The correct answer is: Exactly 1 solution

When two lines intersect at a point on a graph, there is precisely one solution, which corresponds to the coordinates of the point where they cross. This single point represents the unique set of values for the variables involved that satisfy both equations of the lines. In the context of linear equations, if you have two different lines (with different slopes), their intersection indicates that they share a particular solution where both lines are true simultaneously. Therefore, the intersection represents the values that solve both equations, confirming that there is just one solution—the point of intersection. Other scenarios demonstrate different numbers of solutions: if the lines are parallel, there would be no solutions since they never meet. If the two lines are identical, they would lie on top of each other, resulting in infinitely many solutions since any point along the line satisfies both equations. Meanwhile, having two solutions would involve scenarios where the equations are non-linear or intersecting under specific conditions, which isn't applicable for two distinct linear equations intersecting at a single point. Thus, the presence of a single intersection point directly correlates to having one unique solution.