Is it possible to convert from an explicit equation back to its implicit equation? Suppose I have the following explicit equation: \[\left \{ (\frac{1}{2}-\frac{3}{2}t, -\frac{1}{2}+\frac{1}{2}t,t)|t \in \mathbb{R}\right \}\] Possible to change this back to its implicit form?

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Do you, mean for example, as z = t, then x = 1/2 ( 1 - 3z) y = 1/2 ( z - 1) ?

oh..I meant the above explicit form of the equation was derived from the system of equations (implicit form): \[x+y+z=0\]\[x-y+2z=1\] So after getting the explicit form, can I change the explicit form back to this implicit form?

But how do I find the equations of the planes? Is there a systematic way to do this? Probably a linear algebra way? I tried to solve for its nullspace and all of that but none of the ways lead me back to the implicit form of the 2 equations.

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