Both equations are technically correct. When you're using "c" in the first equation, that just means that you're adding some type of constant number. However, the letter "b" is sometimes used to identify the y-intercept of a linear line. But don't get me wrong, "c" can also be used as the y-intercept since it's a constant added number. Therefore, we can conclude that they are THE SAME so therefore both should be accepted by your teacher.
I hope that helps you out! Please let me know if you have any other questions!
College Calculus Student ; Math Tutor on AIM