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For example, is there any difference between the following sentences:
With your ex out of the picture, we can finally start dating. With your ex out of the equation, we can finally start dating.
In general, the only difference I could find is that "out of the picture" seems more common. Apart from this, is there a difference in their meaning too?
"Out of the picture" means that something is no longer around. If someone is literally "in the picture", they are physically present.
"Out of the equation" means that they are no longer having an effect. Literally, each variable in a formula affects the value of that formula.
In many cases the meaning is close enough that the choice of metaphor is up to the speaker, because the two metaphors are so close. If your ex is "in the picture" (ie still present in someone's life) then they must be "in the equation" (having an effect). If they are "out of the picture" then they must be "out the equation". So you could use either.
