When will a system of two linear equations be inconsistent? Select all that apply.

Question

A.
When the graphs of the two equations are parallel

B.
When the graphs of the two equations are perpendicular

C.
When the slopes of the two equations are equal

D.
When the slopes of the two equations are negative reciprocals

E.
When the graphs of the two equations are neither parallel nor perpendicular

Answer 1

A system of two linear equations is considered inconsistent if there are no solutions, meaning the lines representing the equations do not intersect. This occurs in the following situations:

A. When the graphs of the two equations are parallel - This is true; parallel lines do not intersect and therefore the system is inconsistent.

C. When the slopes of the two equations are equal - This is also true; if the slopes are equal and the lines have different y-intercepts, they will be parallel and thus inconsistent.

The other options do not lead to inconsistency:

B. When the graphs of the two equations are perpendicular - This is false; perpendicular lines intersect at one point, meaning the system will have one solution.

D. When the slopes of the two equations are negative reciprocals - This is also false; lines with slopes that are negative reciprocals are perpendicular and will intersect.

E. When the graphs of the two equations are neither parallel nor perpendicular - This statement is false; if the lines are neither parallel nor perpendicular, they will typically intersect at some point, leading to a consistent system.

Thus, the correct answers are:

• A (When the graphs of the two equations are parallel)
• C (When the slopes of the two equations are equal)