Question

questions:

1. if a system of linear equations has one solution, what does this mean about the two lines?
2. if a system of linear equations has no solution, what does this mean about the two lines?
3. if a system of linear equations has infinitely many solutions, what does this mean about the two lines?

Explanation:

1)

Step1: Recall linear system solutions

A system of linear equations represents two lines. One solution means they intersect at one point.

Step2: Relate to line behavior

If two lines intersect at exactly one point, they are not parallel (different slopes) and not coinciding. So they are intersecting (at one point).

Step1: Recall no - solution condition

A system with no solution has equations that represent lines with the same slope but different y - intercepts.

Step2: Determine line relationship

Lines with the same slope and different y - intercepts are parallel (never intersect).

Step1: Recall infinite solutions condition

A system with infinitely many solutions has equations that represent the same line (same slope and same y - intercept).

Step2: Determine line relationship

When two lines are the same, they are coinciding (one line lies exactly on top of the other).

Answer:

The two lines intersect at exactly one point (they have different slopes, are not parallel or coinciding).

2)