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
$\left[A:B\right]=\left[\begin{array}{cccc}1& 2& 3& 1\\ 2& 1& 3& 2\\ 5& 5& 9& 4\end{array}\right]$
Apply $=\{\begin{array}{ll}{R}_2\to R_2-2R_1& \\ R_3\to R_3-5R_1& \end{array}$
$\sim \left[\begin{array}{cccc}1& 2& 3& 1\\ 0& -3& -3& 0\\ 0& -5& -6& -1\end{array}\right]$
Apply $R_3\to R_3-\frac{5}{3}{R}_2$
$\sim \left[\begin{array}{cccc}1& 2& 3& 1\\ 0& -3& -3& 0\\ 0& 0& -1& -1\end{array}\right]$
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