Two equations describe the same line. How many distinct solutions exist?

Two equations describe the same line. How many distinct solutions exist?

• A. Infinitely many solutions
• B. No solution
• C. One solution
• D. The lines cross at a point

Answer: (A)

When two equations describe the same line, every point on that line satisfies both equations, so there are infinitely many solutions. The system is dependent, meaning one equation can be rearranged or scaled to look like the other, and no single point is singled out. For example, if one equation is y = 2x + 1 and the other is 4x − 2y + 2 = 0, they describe the same line; substituting y from the first into the second reduces to 0 = 0 for all x, so every x gives a valid point on the line. That means an endless set of solutions. By contrast, if the lines are parallel but distinct, there are no common points and thus no solution; if the lines intersect, there is exactly one common point, giving a single solution.

When two equations describe the same line, every point on that line satisfies both equations, so there are infinitely many solutions. The system is dependent, meaning one equation can be rearranged or scaled to look like the other, and no single point is singled out. For example, if one equation is y = 2x + 1 and the other is 4x − 2y + 2 = 0, they describe the same line; substituting y from the first into the second reduces to 0 = 0 for all x, so every x gives a valid point on the line. That means an endless set of solutions. By contrast, if the lines are parallel but distinct, there are no common points and thus no solution; if the lines intersect, there is exactly one common point, giving a single solution.