What are the characteristics of consistent- independent system of linear equations in two variables ? 1. the slopes are different II. infinitely many solutions Hl. no solutions IV. the lines intersect V. the lines are parallel VI. the system has one solution VII. the lines are overlapping A. I, IV, VI B. I, II, II C. I, III, IV D. V, VI, VII
The Correct Answer and Explanation is:
✅ Correct Answer: A. I, IV, VI
📘 Explanation
In the study of systems of linear equations in two variables, we classify systems based on the number of solutions they have. A consistent and independent system is a system of equations that has exactly one solution. Let’s break this down using the characteristics given:
What does consistent mean?
A consistent system has at least one solution. This means the lines either intersect at one point or are the same line (overlapping).
What does independent mean?
An independent system means the equations represent two distinct lines. This excludes overlapping lines, which are considered dependent.
Therefore, a consistent and independent system has the following characteristics:
- I. The slopes are different
✔️ True. Different slopes mean the lines are not parallel. They will intersect at exactly one point, which gives one unique solution. - IV. The lines intersect
✔️ True. If the lines intersect at a single point, the system is consistent (one solution) and independent (lines are distinct). - VI. The system has one solution
✔️ True. That is the defining feature of a consistent and independent system.
Why are the other options incorrect?
- II. Infinitely many solutions ❌ → This describes a dependent system, not independent.
- III. No solutions ❌ → This applies to an inconsistent system where the lines are parallel.
- V. The lines are parallel ❌ → Parallel lines never intersect, so the system is inconsistent.
- VII. The lines are overlapping ❌ → This is a consistent but dependent system.
✅ Final Answer: A. I, IV, VI
This system represents two lines with different slopes that intersect at one unique point, meaning the system is consistent and independent.
