Question

if two equations represent the same line, the system has solutions.

Explanation:

Understand systems of equations

Using the System of Equations Definition knowledge point
A system of linear equations consists of two or more linear equations. A solution to the system is any coordinate point \((x, y)\) that satisfies all equations in the system simultaneously.

Analyze the graphical representation

Using the Graphical Solution of Systems knowledge point
Graphically, the solutions to a system of linear equations correspond to the points where the graphs of the equations intersect.

• If two lines intersect at exactly one point, the system has one unique solution.
• If two lines are parallel and never intersect, the system has no solution.
• If two equations represent the same line, they lie directly on top of each other (coincident lines). Every point on the line is a point of intersection, meaning they share infinitely many points. Therefore, the system has infinitely many solutions.

Answer:

If two equations represent the same line, the system has <blank>infinitely many</blank> solutions.