Hello, I have learned that the most distant galaxies accessible to us by telescope are about 14 billion light years away. Accordingly, the method used to determine the distance of said galaxies is to use the old light of supernovae that existed within those galaxies. These exploding stars mark the end of a stars long life, but the explosions themselves occurred some 14 billion years ago. My question is that if what I have explained is true, how were those stars formed before the big bang (13.7 billion years ago). Were they already formed in the hot condensed state of the singularity that has now come to be the universe?
Your question is a very insightful one. Normally if you want to know how long it takes for something to travel you can use the formula: velocity=distance/time or equivalently time=distance/velocity.
Using this formula on a distant galaxy you would arrive at the conclusion that you mentioned in your question.
However you have to keep in mind that the distant galaxy that emitted the light is not sitting still relative to us. In fact the space between us and that galaxy is expanding. As a picture to have in your mind: imagine that the universe is a loaf of raisin bread. Each raisin in the bread in this model represents a galaxy. Now imagine that you bake the bread and you put it in an oven. As the bread heats up, the yeast in the bread causes it to expand. Any two raisin in the bread are now moving away from each other in the bread.
Thus the reason why it is more distant than (the speed of light)x(age of the universe) is because it has been moving away from us since the light that we see was emitted.
I hope that answers your question. Feel free to shoot me back an email if you need more clarification.
Robert da Silva