Does come through with sound like a perfectly acceptable idiomatic alternative to come up with?
"He came through with an answer, not immediately, that made so much sense." – source
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Only in some instances could they be used interchangeably. Although they are similar in meaning, "come through with" means to deliver whereas "come up with" means to find.
Johnny came up with $10. Johnny found $10 (Johnny is the recipient).
Johnny came through with $10. Johnny delivered $10 (Johnny is the deliverer).
However, if everybody in a group has a goal that requires $10 more to accomplish, then you could say either one to describe the fact that Johnny solved the problem with $10.
On paper, the definitions looks similar:
come through: 1. to do what is needed or expected [came through in the clutch]
come up with: to produce especially in dealing with a problem or challenge [came up with a solution]
In informal American English, however, come through carries a sense of avoiding some negative consequence with the action. If my co-worker's presentation saves a failing sales pitch, I might remark
Sally really came through for us with her slides.
Mark arrived late, but he came through during the Q&A.
The focus is on the salutary effect of the act.
On the other hand, to come up with something suggests an invention or an ad hoc solution. It can have both positive or negative connotations depending on context:
The Winkelvosses came up with the idea for Facebook— or so they claim.
Whoever came up with turbaconducken was a genius.
Perhaps the thing come up with solves a pressing problem, and perhaps not, but it is in any case something newly devised. The focus is on innovative nature of the act.
Thus, there are situations where either could be used, but they would highlight slightly different aspects.
The application was bogging down, but the DBA came through with a more efficient stored procedure.
The DBA found a solution, to everyone's relief.
The application was bogging down, but the DBA came up with a more efficient stored procedure.
The DBA found an inventive solution.