I was trying to solve one question which is asking to find a plane which passes through given point and is parallel to given line.
The given point is $M(2,-5,3)$ and the given line is given as an interesection of the planes $2x-y+3z-1=0 \text{ and } 5x+4y-z-7=0$
It is still unclear for me why there is only one unique plane which can be answer, I think that there are more possible planes that can be answers to this.
Best Answer
Your are right, such plane is not unique. For example the planes $2x-y+3z=18$ and $5x+4y-z=-13$ pass through the point $(2,-5,3)$ and they are parallel to the given line.
More generally, through the given point, there is a unique line parallel to the given line, but then any plane through this second line is parallel to the given line.