Ohio Assessments for Educators (OAE) Mathematics Practice Exam

What does it mean if two equations are coincident?

1. They intersect at two points

2. They have no solutions

3. They have an infinite number of solutions

4. They intersect at one point

The correct answer is: They have an infinite number of solutions

When two equations are described as coincident, it means they represent the same line in a coordinate system. This occurs when both equations produce identical solutions for every value of the variable involved. As a result, there are infinitely many points that satisfy both equations, hence the statement that they have an infinite number of solutions is accurate. In a geometric context, this is visualized as two lines lying exactly on top of each other, making every point on the line a solution to both equations. In contrast, the other options do not accurately reflect the nature of coincident lines: intersecting at two points implies distinct equations whose graphs cross, having no solutions indicates the lines are parallel and never meet, and intersecting at one point suggests the equations are distinct and have a unique solution. These alternatives do not capture the essence of coincident lines, which is their complete overlap.