To show whether or not the 3 planes
$$x+y-2z=5\tag 1$$
$$x-y+3z=6 \tag2$$
$$x+5y-12z=12 \tag 3$$ all have a common line of intersection.
Can I do $(3)-(2)$ to get the line $6y-15z=6$ and $(1)-(2)$ to get the line $2y-5z=-1$ which is $6y-15z=-3$ , and say that as these aren't the same line, they don't have a common line of intersection?
Another thing that is confusing me is that if instead of eliminating $x$, I chose to eliminate $z$, I would get different lines in terms of $x$ and $y$. But how can I get the equations of two different lines by eliminating from the same pair of plane equations? There's only one line of intersection between any pair of planes, so surely I should only be able to get one unique line if I eliminate a variable from a pair of planes?
Any help would be appreciated
Best Answer
An elementary solution and notes to the OP
If $3$ planes have a unique common point then they don't have a common straight line. In order to see if there is a common line we have to see if we can solve the following system of equations:
$$ \begin{matrix} x+y&-2z&=&5\\ x-y&+3z&=&6\\ \ x+5y&-12z&=&12. \end{matrix} $$ Adding the first equation to the second one we get $$2x+z=11.$$ Multiplying the second equation by $5$ and then adding it to the third equation we get $$3x+z=21.$$ These two equations have a unique solution: $$x=10\text { and } z=-9.$$ Substituting these numbers back to any of the original equations we get $y=-23$.
So, the three planes have a unique common point; no common line exists.
Notes to the OP
If you take, say, $(1)$ and $(2)$ and eliminate one of the variables, say $x$ then you get an equation of a straight line in the plane $zy$.
This line is a perpendicular projection of the common line of $(1)$ and $(2)$ to $yz$. If the planes $(1)$, $(2)$, and $(3)$ have a unique point then all of the possible eliminations will result in a triplet of straight lines in the different coordinate planes.
By erecting a perpendiculars from the common points of the said line triplets you will get back to the common point of the three planes.
I hope that this brief explanation helped you to understand better your own efforts.