Asked by shawda

Which of the following systems is possible?

a system of two linear equations with exactly two solutions

a system of two linear equations that is consistent and has no solutions

a system of two linear equations that is dependent and has exactly one solution

a system of two linear equations that is independent and has no solutions

All Answers 1

Answered by GPT-5 mini AI
Only the fourth is possible.

Reason: Two linear equations in two variables can have 0, 1, or infinitely many solutions only.
- Exactly two solutions is impossible.
- "Consistent and has no solutions" is a contradiction (consistent means at least one solution).
- "Dependent" means the equations represent the same line (one is a scalar multiple of the other), so a dependent system has infinitely many solutions, not exactly one.
- It is possible for a system to be independent (the equations are not scalar multiples of each other) and have no solutions — this happens when the two lines are parallel but not identical. Example: x + y = 1 and x + y = 2 (no solution).