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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.

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  • $\begingroup$ Perhaps you might include the specific problem that you’re asking about. $\endgroup$ – amd Mar 13 at 19:40
  • $\begingroup$ I inserted the given point and the line into the post $\endgroup$ – someone123123 Mar 13 at 20:30
  • $\begingroup$ Are you sure that the problem said for the plane to be parallel to that line? If it must instead include the line, then the solution is be unique. $\endgroup$ – amd Mar 13 at 20:35
  • $\begingroup$ Yes, the question is asking about plane which is parallel to the given line. $\endgroup$ – someone123123 Mar 13 at 20:45
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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.

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  • $\begingroup$ I'm thinking like this, but my teacher gave some explanation that I didn't really understand. She says that there is only one such plane $\endgroup$ – someone123123 Mar 13 at 9:57
  • $\begingroup$ Maybe the problem was "find a plane which passes through a given point and it is orthogonal to a given line. $\endgroup$ – Robert Z Mar 13 at 10:00
  • $\begingroup$ Or perhaps the plane is supposed to include the intersection line of the two given planes (see updated question). Impossible to know without seeing the teacher’s solution. $\endgroup$ – amd Mar 14 at 0:24
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You are right: there are infinitely mane planes passing through a point and parallel to a given line.

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As another answer points out, the claim is false. Given a set of planes parallel to each other as well as to the given line, only one of those planes will pass through the given point.

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