Question

The values of $a$ and $b$, for which the system of equations
$2x+3y+6z=8$
$x+2y+az=5$
$3x+5y+9z=b$
has no solution, are

• A. $a=3,b=13$
• B. $a=3,b\ne 13$
• C. $a\ne 3,b\ne 13$
• D. $a\ne 3,b=3$

Answer 1

The correct option is B $a=3,b\ne 13$

For no solution $\Delta =0$ (by Cramer’s rule)
$\Rightarrow \left|\begin{array}{ccc}2& 3& 6\\ 1& 2& a\\ 3& 5& 9\end{array}\right|=0$
$\Rightarrow 2(18-5a)-3(9-3a)+6(-1)=0$
$\Rightarrow -a+3=0\Rightarrow a=3$
Let solution of $2x+3y+6z=8$ and $x+2y+3z=5$ be $z=k$ gives $y=2$ and $x=1–3k$
for no solution $(1-3k,2,k)$ shall not satisfy $3x+5y+9z=b$
$\therefore 3(1-3k)+10+9k\ne b$
$\Rightarrow b\ne 13$