1. **Stating the problem:** We need to discuss the nature of solutions of systems of linear equations, specifically the types: consistent, inconsistent, independent, and dependent. 2. **Consistent system:** A system is consistent if it has at least one solution. Example: Consider the system $$\begin{cases} x + y = 2 \\ x - y = 0 \end{cases}$$ Adding the two equations gives $2x = 2 \Rightarrow x = 1$, then $y = 1$. So the system has a unique solution $(1,1)$. 3. **Inconsistent system:** A system is inconsistent if it has no solution. Example: Consider $$\begin{cases} x + y = 2 \\ x + y = 3 \end{cases}$$ These two equations contradict each other because the same left side equals two different right sides. Hence, no solution exists. 4. **Independent system:** A consistent system with exactly one unique solution is called independent. Example: The first example above is independent because it has a unique solution. 5. **Dependent system:** A consistent system with infinitely many solutions is called dependent. Example: Consider $$\begin{cases} x + y = 2 \\ 2x + 2y = 4 \end{cases}$$ The second equation is just twice the first, so they represent the same line. Hence, infinitely many solutions exist along the line $x + y = 2$. **Summary:** - Consistent: at least one solution (includes independent and dependent). - Inconsistent: no solution. - Independent: exactly one solution. - Dependent: infinitely many solutions.