Question

For the equations $ x + 2y + 3z = 1 $ , $ 2x + y + 3z = 2 $ , $ 5x + 5y + 9z = 4 $

• A. there is only one solution
• B. there exists infinitely many solutions
• C. there is no solution
• D. None of the above

Answer 1

Answer: (A)

Given system of equation is
$x + 2y + 3z = 1$
$2x + y + 3z = 2$
$5x + 5y + 9z = 4$
The augmented matrix
$\left[A : B\right]=\left[\begin{matrix}1&2&3&1\\ 2&1&3&2\\ 5&5&9&4\end{matrix}\right]$
Apply $ = \begin{cases} R_{2}\to R_{2}-2R_{1} & \text{} \\ R_{3}\to R_{3}-5R_{1} & \text{} \end{cases}$
$\sim\left[\begin{matrix}1&2&3&1\\ 0&-3&-3&0\\ 0&-5&-6&-1\end{matrix}\right]$
Apply $R_{3}\rightarrow R_{3}-\frac{5}{3}R_{2}$
$\sim\left[\begin{matrix}1&2&3&1\\ 0&-3&-3&0\\ 0&0&-1&-1\end{matrix}\right]$
Here, rank of $\left[A:B\right]$ =rank of $\left[A\right]$
So, the system is consistent
But, rank of $ \left[A\right]$ = number of unknowns
Hence, the system have only one solution