Question

Which of the following systems of linear equations has:

(i) a unique solution?
(ii) infinitely many solutions?
(iii) no solution?

For each case, find all solutions and provide a detailed explanation of the method or technique used to determine the nature of the solution (unique, infinite, or none). Include relevant mathematical expressions and any critical points involved in the analysis.

Answer 1

To determine the nature of the solution for a system of linear equations, we need to solve the equations and analyze the results.

Step-by-step explanation:

To determine the nature of the solutions, we need to solve each system of linear equations and analyze the results.

1. If a system has a unique solution, it means there is exactly one set of values for the variables that satisfies both equations. This occurs when the two lines representing the equations intersect at a single point.
2. If a system has infinitely many solutions, it means all points on one line are also on the other line. In other words, the two lines are the same.
3. If a system has no solution, it means the lines representing the equations are parallel and will never intersect.

By solving each system of equations, we can determine which category each falls into.