A system of two linear equations in two variables has no solution. What can be said about the lines?

Question

Accepted Answer

The lines are parallel but not identical. A system of two linear equations with no solution is called inconsistent. Geometrically, this means the two lines never intersect, which occurs when the lines are parallel but not identical. Parallel lines have the same slope but different y-intercepts, so they never cross.

If the lines intersected at exactly one point ("The lines intersect at exactly one point"), the system would have exactly one solution  -  a consistent and independent system. If the lines were identical ("The lines are identical"), every point on the line would be a solution, giving infinitely many solutions  -  a consistent and dependent system. Perpendicular lines ("The lines are perpendicular") always intersect at exactly one point (at a 90-degree angle), so they would produce exactly one solution.

Understanding the geometric interpretation of systems is important for the CLEP exam. To check if lines are parallel algebraically, rewrite both equations in slope-intercept form (y = mx + b) and compare slopes. If the slopes are equal but the y-intercepts differ, the lines are parallel and the system has no solution. If the slopes are equal and the y-intercepts are also equal, the lines are identical and the system has infinitely many solutions.

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