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Do we write

  • the relationship recast, OR
  • the relationship recasts? ("s" at the end)

y = 3x       (1)

Relationship (1) recast

y−3x = 0  (2)

I have many times encountered the verb without -s at the end.

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In this, recast is a past participle: it is equivalent to saying rewritten, or re-expressed.

Recasts would be odd, because the equation is the object, not the subject of the recasting.

It's confusing because cast and its compounds are among the few words where the past participle is the same as the present. Hit is another case.

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