1. **Problem Statement:** Determine the nature of the system of linear equations: $$\begin{cases} x + y + z = 6 \\ x + 2y + 3z = 14 \\ x + 4y + 7z = 30 \end{cases}$$ 2. **Formula and Rules:** - The system is consistent if the rank of the coefficient matrix equals the rank of the augmented matrix. - If this rank is equal to the number of variables, the system has a unique solution. - If the rank is less than the number of variables, the system has infinitely many solutions. - If the ranks differ, the system is inconsistent. 3. **Step-by-step Solution:** - Coefficient matrix $A = \begin{bmatrix}1 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 4 & 7\end{bmatrix}$ - Augmented matrix $[A|b] = \begin{bmatrix}1 & 1 & 1 & 6 \\ 1 & 2 & 3 & 14 \\ 1 & 4 & 7 & 30\end{bmatrix}$ - Calculate rank of $A$: - Subtract row 1 from rows 2 and 3: $$\begin{bmatrix}1 & 1 & 1 \\ 0 & 1 & 2 \\ 0 & 3 & 6\end{bmatrix}$$ - Subtract 3 times row 2 from row 3: $$\begin{bmatrix}1 & 1 & 1 \\ 0 & 1 & 2 \\ 0 & 0 & 0\end{bmatrix}$$ - Rank of $A$ is 2 (two non-zero rows). - Calculate rank of augmented matrix $[A|b]$ similarly: - Subtract row 1 from rows 2 and 3: $$\begin{bmatrix}1 & 1 & 1 & 6 \\ 0 & 1 & 2 & 8 \\ 0 & 3 & 6 & 24\end{bmatrix}$$ - Subtract 3 times row 2 from row 3: $$\begin{bmatrix}1 & 1 & 1 & 6 \\ 0 & 1 & 2 & 8 \\ 0 & 0 & 0 & 0\end{bmatrix}$$ - Rank of augmented matrix is also 2. - Number of variables is 3. 4. **Conclusion:** - Since rank($A$) = rank($[A|b]$) = 2 < 3 (number of variables), the system is consistent and has infinitely many solutions. **Final answer:** b) Consistent and has infinite solutions.