Anew machine can produce two more engine parts in an hour than an old machine. therefore, the new machine finished a certain job in 7 hours while the old one needed 8 hours to produce the same number of engine parts. how many parts did the new machine produce in an hour?

For this type of problems, we try to work with the rate of production per unit time.

Let
x=rate of production of old machine (parts/hour)
x+2=rate of production of new machine  (parts/hour)

"the new machine finished a certain job in 7 hours while the old one needed 8 hours to produce the same number of engine parts. "  
means
7(x+2) = 8x
Solve for x
8x-7x=14
x=14 parts/hour (rate of old machine)
x+2=16 parts/hour (rate of new machine)

Check:  7(14+2)=112, 8(14)=112  ✔

Wow. that is a really cool question.