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I know that there are similar questions to this on stackexchange, but I really don't understand the answers/explanations there.

How would you parametrize an equation of a hyperplane in $\mathbb{R}^4$ given by the equation $$ 2x + y + z + 3t = 5. $$ Can I just assume y, z, and t are free variables and write...

$$ \begin{split} x &= (5-p-q-3r)/2\\ y &=p\\ z &=q\\ t &=r \end{split} $$

or is this not correct? If not correct then why is it not?

Thanks!

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    Correct. What do you want it for? – 2017-01-23
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    Hm, it's a question from my textbook but I was wondering if I can always just assume that y,z,t are free variables and show x in terms of y,z,t. – 2017-01-23

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