What is the System of Equations for Finding Planes Containing a Given Line?

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To find a system of equations for planes containing the line defined by \(\frac{x-1}{2}=\frac{y+2}{-3}=\frac{z-3}{4}\), one equation derived is \(a - 2b + 3c = 1\). To create a second equation, two points on the line, such as (1, -2, 3) and another point derived from the line's parametric form, should be used. Plugging these points into the plane equation \(ax + by + cz = 1\) will yield the necessary equations. The discussion emphasizes the importance of selecting a second point to complete the system. The goal is to find all planes that include the specified line.
jdstokes
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Let \ell be the line given by \frac{x-1}{2}=\frac{y+2}{-3}=\frac{z-3}{4}. Write down a system of two equations in the three unknowns a, b, c whose solutions give all planes ax + by + cz = 1 in which \ell lies, and solve the system. I can certainly write that a -2b + 3c = 1. I can't figure out how to obtain the second equation.

Thanks

James.
 
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Just pick two points on the line and plug in for (x,y,z).
 
I picked (1,-2,3) and I only have one equation a -2b + 3c = 1.
 
so pick another one...
 

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