x + 2y + 3z = \alpha< / p > < p > </p><p> < / p >< p > < / p > < p > </p><p> < / p >< p > < / p > < p > T h e s y s t e m o f l i n e a r e q u a t i o n s i s c o n s i s t e n t i t m e a n s i t h a s u n i q u e s o l u t i o n o r i n f i n i t e s o l u t i o n s < / p > < p > H e r e , </p><p>The system of linear equations is consistent it means it has unique solution or infinite solutions</p><p>Here, < / p >< p > T h esys t e m o f l in e a re q u a t i o n s i sco n s i s t e n t i t m e an s i t ha s u ni q u eso l u t i o n or in f ini t eso l u t i o n s < / p >< p > Here , { 1 2 3 4 5 6 7 8 9 \begin{cases}1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9\end{cases} ⎩ ⎨ ⎧ 1 4 7 2 5 8 3 6 9 < / p > < p > H e n c e , t h e r e a r e i n f i n i t e l y m a n y s o l u t i o n s . < / p > < p > I f e q u a t i o n s h a v e i n f i n i t e l y m a n y s o l u t i o n s , t h e n t h e e q u a t i o n s a r e l i n e a r l y c o n n e c t e d , i . e . , </p><p>Hence, there are infinitely many solutions.</p><p>If equations have infinitely many solutions, then the equations are linearly connected, i.e., < / p >< p > He n ce , t h ere a re in f ini t e l y man yso l u t i o n s . < / p >< p > I f e q u a t i o n s ha v e in f ini t e l y man yso l u t i o n s , t h e n t h ee q u a t i o n s a re l in e a r l yco nn ec t e d , i . e . , < / p > < p > </p><p> < / p >< p > < / p > < p > </p><p> < / p >< p > < / p > < p > </p><p> < / p >< p > < / p > < p > A l s o , </p><p>Also, < / p >< p > A l so , < / p > < p > </p><p> < / p >< p > < / p > < p > </p><p> < / p >< p > < / p > < p > N o w , </p><p>Now, < / p >< p > N o w , i s t h e p l a n e c o n t a i n i n g t h e p o i n t s ( α , β , γ ) is the plane containing the points \left(\alpha, \beta, \gamma\right) i s t h e pl an eco n t ainin g t h e p o in t s ( α , β , γ )
So, the equation of the plane is x − 2 y + z = 1 x - 2y + z = 1 x − 2 y + z = 1 α , β , γ \alpha, \beta, \gamma α , β , γ x , y , z x, y, z x , y , z
We know, the distance of a point \left({x}_{1}, {y}_{1}, {z}_{1}\right)f r o m p l a n e from plane f ro m pl an e i s ∣ a x 1 + b y 1 + c z 1 + d a 2 + b 2 + c 2 ∣ is \left|\dfrac{a{x}_{1} + b{y}_{1} + c{z}_{1} + d}{\sqrt{{a}^{2} + {b}^{2} + {c}^{2}}}\right| i s a 2 + b 2 + c 2 a x 1 + b y 1 + c z 1 + d
So, D = ( 0 × 1 − 2 × 1 + 0 × 1 − 1 1 2 + ( − 2 ) 2 + 1 2 ) 2 = 9 6 = 1.50 D = {\left(\dfrac{0 \times 1 - 2 \times 1 + 0 \times 1 - 1}{\sqrt{{1}^{2} + ( - 2{)}^{2} + {1}^{2}}}\right)}^{2} = \dfrac{9}{6} = 1 . 50 D = ( 1 2 + ( − 2 ) 2 + 1 2 0 × 1 − 2 × 1 + 0 × 1 − 1 ) 2 = 6 9 = 1.50
