For a system of three equations in two unknowns to have no solutions, the lines must be positioned in such a way that there is no point of intersection common to all three lines. Let's examine the provided options:

1. The three lines intersect in a single point.

   • This situation would result in a solution since all three lines meet at a common point.
   • Not applicable for no solutions.

2. The lines are all parallel.

   • If the lines are all parallel, they will never intersect, which implies there are no common points of intersection.
   • This is a valid case for no solutions.

3. Two of the lines are parallel.

   • If two lines are parallel, but the third line intersects them at different points, there is no single point where all three lines meet.
   • This is a valid case for no solutions.

4. Two of the lines are the same, and the third line intersects it in a single point.

   • In this case, two of the lines are identical, and the third intersects them at one point. This scenario results in at least one point of intersection, meaning there is a solution.
   • Not applicable for no solutions.

5. The three lines intersect in three different points (forming a triangle).

   • If the three lines intersect in three different points, there is no common point where all three lines meet, meaning no solutions exist.
   • This is a valid case for no solutions.

Valid scenarios where the system has no solutions:

• The lines are all parallel.
• Two of the lines are parallel.
• The three lines intersect in three different points (forming a triangle).

Do you have any questions or would you like more details on any of these?

Here are five related questions:

1. What happens if all three lines coincide?
2. How does the number of solutions change if the lines intersect in two different points?
3. What is the geometric interpretation of systems of equations with exactly one solution?
4. How would the system behave if two lines were parallel, but the third was coincident with one of the parallel lines?
5. How would you solve such a system algebraically to determine if it has no solutions?

Tip: Always check the slopes of the lines in a system of linear equations to understand their geometric relationship. Parallel lines have equal slopes!