Question

The existence of the unique solution of the system of equations
$ x + y + z = \beta $
$ 5x - y + \alpha z = 10 $
$ 2x + 3y - z = 6 $
depends on

• A. $ \alpha $ only
• B. $ \beta $ only
• C. $ \alpha $ and $ \beta $ both
• D. neither $ \beta $ nor $ \alpha $

Answer 1

Answer: (A)

Given, system of equation is $x + y + z= \beta$
$5x - y + \alpha z = 10$ and $2x + 3y - z =6$
For unique solution $\left|\begin{matrix}1&1&1\\ 5&-1&\alpha\\ 2&3&-1\end{matrix}\right|\ne0 $
$\Rightarrow 1\left(1-3\alpha\right)-1\left(-5-2\alpha\right)+1\left(15+2\right)\ne0$
$\Rightarrow 1-3\alpha+5+2\alpha+17\ne0$
$\Rightarrow -\alpha+23 \ne0 $
$\Rightarrow \alpha \ne23$
Hence, for the existence of the unique solutionthe system of equations depend on $\alpha$ only