Question

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

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

Answer 1

Answer: (A)

$\alpha$ only

Answer 2

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