1. The problem asks to discuss the nature of solutions of systems of linear equations, specifically consistent, inconsistent, independent, and dependent systems. 2. A system of linear equations can have different types of solutions based on the relationships between the equations. 3. **Consistent system:** A system is consistent if it has at least one solution. This means the equations do not contradict each other. 4. **Example of consistent system:** $$\begin{cases} x + y = 2 \\ 2x - y = 1 \end{cases}$$ This system has a unique solution. 5. **Inconsistent system:** A system is inconsistent if it has no solution. This happens when the equations contradict each other. 6. **Example of inconsistent system:** $$\begin{cases} x + y = 2 \\ x + y = 3 \end{cases}$$ These two lines are parallel and never intersect, so no solution exists. 7. **Independent system:** A system is independent if it has exactly one unique solution. The equations represent different lines that intersect at one point. 8. **Example of independent system:** $$\begin{cases} x + y = 2 \\ 2x - y = 1 \end{cases}$$ Solving gives a unique solution. 9. **Dependent system:** A system is dependent if it has infinitely many solutions. This occurs when the equations represent the same line. 10. **Example of dependent system:** $$\begin{cases} x + y = 2 \\ 2x + 2y = 4 \end{cases}$$ The second equation is just twice the first, so they represent the same line. 11. To summarize: - Consistent means at least one solution (includes independent and dependent). - Inconsistent means no solution. - Independent means exactly one solution. - Dependent means infinitely many solutions. 12. Understanding these types helps in solving and interpreting systems of linear equations in algebra and applied contexts. Word count: 280