This question is opening a can of worms for the PB, judging from recent threads. So, rather than get in a protracted debate with those who disagree with me, I'll just offer this as my

.
Sometimes the NT writers quoted. Sometimes the paraphrased. Sometimes they used the MT, and sometimes they used the LXX. I do not believe the LXX is inspired. However, as any translation is an interpretation, sometimes it seems the NT writers regarded the LXX interpretation as the correct one, and so used it. Other times, perhaps, they did not see a substantial difference (touching their point in the context), and since they were writing in Greek, freely used the Greek translation.
You should understand, however, that there really is no such thing as the LXX. Simply peruse the first chapters of Swete or Jellicoe or any other Septuagint introduction and you'll see that the text critical situation of the LXX is a mess. Add to that the whole issue of Daniel and Theodotian and you'll see that the MT isn't a super clean situation either.
I don't think there is any question that the temple would have been using the Hebrew. The Masoretes clearly demonstrate that the vocalic tradition they preserved was remarkably faithful and precise. So, with few exceptions, I think Jesus would have been dealing with essentially the MT.
But the NT authors use the 'LXX' frequently. But the LXX they use may differ from the one we have today. Origen's well intentioned fiasco has makes knowing what the text they had before them looked like very difficult.
Again, I'm not inviting a discussion of preservation, etc. I'm just offering my two cents. If you'd like more specific information about the textual situation of the LXX (or the MT) let me know. Have I answered your basic question?