Topic: Mathematics
The system of equations $x+y+z=6$, $2x+y+z=7$, $x+2y+z=8$ has
- A. A unique solution
- B. No solution
- C. An infinite number of solutions
- D. None of these
Answer: A. A unique solution
Explanation: Forming the augmented matrix $[A:B] = \begin{bmatrix}1&1&1&6\\ 2&1&1&7\\ 1&2&1&8\end{bmatrix}$. Applying row operations $R_{2}\leftarrow R_{2}-2R_{1}$ and $R_{3}\leftarrow R_{3}-R_{1}$ gives $\begin{bmatrix}1&1&1&6\\ 0&-1&-1&-5\\ 0&1&0&2\end{bmatrix}$. Here Rank of A = Rank of AB = Number of variables. Thus, it is a unique solution.