Question

The ordered pair $(a,b)$, for which the system of linear equations
$3x-2y+z=b$
$5x-8y+9z=3$
$2x+y+az=-1$
has no solution, is :

• A. $(3,-\frac{1}{3})$
• B. $(-3,\frac{1}{3})$
• C. $(-3,-\frac{1}{3})$
• D. $(3,\frac{1}{3})$

Answer 1

The correct option is C $(-3,-\frac{1}{3})$

$\left|\begin{array}{ccc}3& -2& 1\\ 5& -8& 9\\ 2& 1& a\end{array}\right|=0$

$\Rightarrow -14a-42=0\Rightarrow a=-3$

Now, $3\left(\text{equation}\right(1\left)\right)-\left(\text{equation}\right(2\left)\right)-2\left(\text{equation}\right(3\left)\right)$ is
$3(3x-2y+z-b)-(5x-8y+9z-3)-2(2x+y+az+1)=0$
$\Rightarrow -3b+3-2=0\Rightarrow b=\frac{1}{3}$
So for no solution, $a=-3$ and $b\ne \frac{1}{3}$