To determine which statements are true for a system of linear equations that has one unique solution, let's analyze each statement step-by-step.
1. The lines overlap completely.
- If the lines overlap completely, it means they are essentially the same line. This would result in infinitely many solutions because every point on one line is also a point on the other line. Therefore, this statement is false for a system with one unique solution.
2. The lines are parallel.
- If the lines are parallel, they never intersect, leading to no solutions. Thus, this statement is also false for a system with one unique solution.
3. The lines intersect at exactly one point.
- For a system of linear equations to have exactly one unique solution, the lines representing these equations must intersect at one and only one point. This means that this statement is true for a system with one unique solution.
4. The equations are independent.
- Independence of the equations ensures that they are not scalar multiples of each other, which means they are not the same line and will intersect at most at one point. Independence is a necessary condition for the existence of a unique solution. Therefore, this statement is true.
Based on the analysis, the true statements for a system of linear equations with one unique solution are:
- The lines intersect at exactly one point.
- The equations are independent.
In other words, the correct answer is the two statements mentioned above.
