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If the system of linear equations

$2x+2y+3z=a$

$3x-y+5z=b$

$x-3y+2z=c$

where a,b,c are non-zero real numbers, has more than one solution, then :
Option 1)

$b-c-a=0$
Option 2)

$b+c-a=0$
Option 3)

$b-c+a=0$
Option 4)

$a+b+c=0$

Answers (1)

best_answer

Non-homogeneous system of linear equation -

$b\neq 0$

- wherein

Solution of a system of equations -

$x_{1},x_{2},\cdot \cdot \cdot ,x_{n}$ satisfy the system of linear equations $Ax=B$

- wherein

Consistent system of linear equation -

If the system of equations has one or more solutions

-

For these 3 equations having more than 1 solution

$\Rightarrow D=0$

$\Rightarrow \begin{vmatrix} 2 &2 &3 \\ 3 & -1 &5 \\ 1&-3 &2 \end{vmatrix}=0$

$\Rightarrow 26-20-24=0$

$\Rightarrow D=0$

Also, $D_{1}=D_2=D_3=0$

$D_1= \begin{vmatrix} a &2 &3 \\ b & -1 &5 \\ c&-3 &2 \end{vmatrix}=0$

$\Rightarrow a\left ( 13 \right )-b\left ( 13 \right )+c\left ( 13 \right )=0$

$\Rightarrow a-b+c=0$

$D_2= 0\Rightarrow \begin{vmatrix} 2 &a &3 \\ 3 & b &5 \\ 1&c &2 \end{vmatrix}=0$

$\Rightarrow a-b+c=0$

$D_3= 0\Rightarrow \begin{vmatrix} 2 &2 &a \\ 3 &-1 &b \\ -1&-3 &c \end{vmatrix}=0$

$\Rightarrow a-b+c=0$

Option 1)

$b-c-a=0$
Option 2)

$b+c-a=0$

Option 3)

$b-c+a=0$

Option 4)

$a+b+c=0$

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