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If the system of equattions

$x+y+z =5$

$x+2y+3z =9$

$x+3y+\alpha z =\beta$

has infinetly many solutions, $\beta -\alpha$ equals:

Option 1)

21

Option 2)

8

Option 3)

18

Option 4)

5

Answers (1)

best_answer

Solution of a non-homogeneous system of linear equations by matrix method -

If $A$ is a singular matrix and $adj(A).b=0$ then the system of equations given by $Ax=b$ has infinitely many solutions or no solution.

- wherein

From the concept for $\infty$ many solution .

$\Delta = \begin{vmatrix} 1 & 1 &1 \\ 1 &2 &3 \\ 1 & 3 & \alpha \end{vmatrix} = \begin{vmatrix} 1 & 1 &1 \\ 0 & 1 & 2\\ 0& 2 & \alpha - 1 \end{vmatrix} =( \alpha - 1) - 4 = \alpha - 5 = 0$

$\alpha = 5$

$\Delta _{1} =\begin{vmatrix} 5 & 1 &1 \\ 9 & 2& 3\\ \beta & 3 & 5 \end{vmatrix} = 0$

$= 2 + \beta -15 = 0 \Rightarrow \beta = 13$

on $\beta = 13$ we get $\Delta _{2} = 0 \and \ \Delta _{3} =0$

$\alpha = 5 , \beta = 13$

$\beta - \alpha = 8$

alpha = 5

Option 1)

21

Option 2)

8

Option 3)

18

Option 4)

5

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