Question

The number of solutions of the triplet $(x,y,z)$ in the following equations are
$2x+4y+6z=8$
$x+2y+3z=5$
$x+y+3z=4$

• A. 0
• B. 1
• C. 3
• D. $\infty$

Answer 1

The correct option is A 0

The given system is
$\left[\begin{array}{ccc}2& 4& 6\\ 1& 2& 3\\ 1& 1& 3\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}8\\ 5\\ 4\end{array}\right]$
or $AX=B$
where
$A=\left[\begin{array}{ccc}2& 4& 6\\ 1& 2& 3\\ 1& 1& 3\end{array}\right],x=\left[\begin{array}{c}x\\ y\\ z\end{array}\right]$ and
$B=\left[\begin{array}{c}8\\ 5\\ 4\end{array}\right]$
Now,
$|A|=\left|\begin{array}{ccc}2& 4& 6\\ 1& 2& 3\\ 1& 1& 3\end{array}\right|$
$R_2\to R_2-R_1/2$
$|A|=\left|\begin{array}{ccc}2& 4& 6\\ 0& 0& 0\\ 1& 1& 3\end{array}\right|=0$
Thus, the system or equations is inconsistent.