1. **Problem statement:** Check if the points $(2, 1, 1)$ and $(\frac{11}{4}, -1, -\frac{3}{4})$ are solutions to the system: $$\begin{cases} 3x + 2y - z = 7 \\ 4x + 9y = 2 \\ x + 5y - 3z = 0 \end{cases}$$ 2. **Check point $(2,1,1)$:** - Substitute into first equation: $3(2) + 2(1) - 1 = 6 + 2 - 1 = 7$ ✔ matches right side - Substitute into second equation: $4(2) + 9(1) = 8 + 9 = 17 \neq 2$ ✘ does not match - Substitute into third equation: $2 + 5(1) - 3(1) = 2 + 5 - 3 = 4$ ✘ does not match So, $(2,1,1)$ **does not solve** the system. 3. **Check point $(\frac{11}{4}, -1, -\frac{3}{4})$:** - First equation: $3(\frac{11}{4}) + 2(-1) - (-\frac{3}{4}) = \frac{33}{4} - 2 + \frac{3}{4} = \frac{33 + 3}{4} - 2 = \frac{36}{4} - 2 = 9 - 2 = 7$ ✔ - Second equation: $4(\frac{11}{4}) + 9(-1) = 11 - 9 = 2$ ✔ - Third equation: $(\frac{11}{4}) + 5(-1) - 3(-\frac{3}{4}) = \frac{11}{4} - 5 + \frac{9}{4} = \frac{11 + 9}{4} - 5 = \frac{20}{4} - 5 = 5 - 5 = 0$ ✔ Thus, $(\frac{11}{4}, -1, -\frac{3}{4})$ **solves** the system. 4. **Second problem:** Determine if the system $$\begin{cases} x + 2y - z = 3 \\ 2x + 4y - 2z = 6 \end{cases}$$ is consistent, inconsistent, or dependent using the rank method. 5. **Form the augmented matrix:** $$\left[\begin{array}{ccc|c}1 & 2 & -1 & 3 \\ 2 & 4 & -2 & 6 \end{array}\right]$$ 6. **Check the rank of coefficient matrix $A$ and augmented matrix $[A|b]$:** - Coefficient matrix $A = \begin{bmatrix}1 & 2 & -1 \\ 2 & 4 & -2 \end{bmatrix}$ - Perform row operations: - $R_2 - 2R_1 \to R_2$ gives $$\begin{bmatrix}1 & 2 & -1 \\ 0 & 0 & 0 \end{bmatrix}$$ - Rank of $A$ is 1 (one nonzero row). - Augmented matrix after same operation: $$\begin{bmatrix}1 & 2 & -1 & 3 \\ 0 & 0 & 0 & 0 \end{bmatrix}$$ - Rank of augmented matrix is also 1. 7. **Conclusion:** Since $rank(A) = rank([A|b]) = 1 < $ number of variables (3), system is **consistent and dependent** (infinite solutions). **Final answers:** - $(2,1,1)$ does not solve the first system. - $(\frac{11}{4}, -1, -\frac{3}{4})$ does solve it. - The second system is consistent and dependent.