Chapter 8.10, Problem 24HP

To determine

The ways to give the relation of two lines or linear equations and the system of equations has exactly two solutions or not is to be explained.

Answer to Problem 24HP

No, a system of linear equations in two variables may have zero, one, or infinitely many solutions.

Explanation of Solution

The two lines or two linear equations may have the zero, one, or infinitely many solutions.

These are three ways to analyse the relation between the two lines:

One solution:

If the two lines are non-parallel and coplanar, then they will intersect in exactly one point which is said to be two lines has single or one solution.

If they are not coplanar, they will have no intersection and therefore no solution.

No solution:

When the two lines do not intersect to each other when they are non-coincident and parallel, there is no solution condition occurs.

Infinite solutions:

When two lines are same or coincident, they intersect at all points along the line, so one can say that there is infinitely solutions are avaible for these two lines.

Conclusion:

Hence, the three ways to give the relation between two lines are one solution, no solution or infinitely solution.