1. **State the problem:** Determine if the system of equations $$\begin{cases} x + 2y = 8 \\ x - y = 4 \end{cases}$$ is consistent or inconsistent and dependent or independent. 2. **Recall definitions:** - A system is **consistent** if it has at least one solution. - It is **inconsistent** if it has no solutions. - It is **dependent** if the equations represent the same line (infinitely many solutions). - It is **independent** if the equations represent different lines that intersect at exactly one point. 3. **Solve the system:** From the second equation, express $x$: $$x = y + 4$$ Substitute into the first equation: $$y + 4 + 2y = 8$$ Simplify: $$3y + 4 = 8$$ $$3y = 4$$ $$y = \frac{4}{3}$$ Find $x$: $$x = \frac{4}{3} + 4 = \frac{4}{3} + \frac{12}{3} = \frac{16}{3}$$ 4. **Interpretation:** The system has a unique solution $\left(\frac{16}{3}, \frac{4}{3}\right)$, so it is **consistent** and **independent**. **Final answer:** The system is consistent and independent.