In a math problem, if a system of equations has no solution, what is the relationship of the lines?

In a math problem, if a system of equations has no solution, what is the relationship of the lines?

• A. They are coincident.
• B. They intersect at one point.
• C. They are parallel.
• D. They are perpendicular.

Answer: (C)

When a system of linear equations has no solution, the two lines corresponding to those equations never meet. That happens when they are parallel: they have the same slope, so they rise and fall at the same rate, but their starting points are different, meaning different intercepts. Because they never cross, there’s no point that satisfies both equations. For example, y = 2x + 1 and y = 2x - 3 share the same slope but different intercepts, so the lines run side by side and never intersect, giving no solution to the system. If the intercepts were the same, they would lie on top of each other (coincident), yielding infinitely many solutions. If the slopes were different, the lines would intersect at exactly one point, giving a single solution. Perpendicular lines also intersect at one point.

When a system of linear equations has no solution, the two lines corresponding to those equations never meet. That happens when they are parallel: they have the same slope, so they rise and fall at the same rate, but their starting points are different, meaning different intercepts. Because they never cross, there’s no point that satisfies both equations.

For example, y = 2x + 1 and y = 2x - 3 share the same slope but different intercepts, so the lines run side by side and never intersect, giving no solution to the system.

If the intercepts were the same, they would lie on top of each other (coincident), yielding infinitely many solutions. If the slopes were different, the lines would intersect at exactly one point, giving a single solution. Perpendicular lines also intersect at one point.