Is there ever a time when the simplest solution won't work?

[Tyrion101]

Whenever I'm on math class I've noticed that often professors, and in my online class the videos and examples all seem to be the most complicated way to finding the answer. For example if solving for an equation by elimination they'll go for eliminating y, even though you could just as easily eliminate x without any modifications to the equation. Do they do this because they just prefer to eliminate y first? My second question is the topic. Is there ever a time when you just have to use that complicated solution your teacher taught you? I use the simplest way to solve a problem to eliminate as many possibilities for error I can. Is this a bad habit?

 

[micromass]

Whenever I'm on math class I've noticed that often professors, and in my online class the videos and examples all seem to be the most complicated way to finding the answer. For example if solving for an equation by elimination they'll go for eliminating y, even though you could just as easily eliminate x without any modifications to the equation. Do they do this because they just prefer to eliminate y first? My second question is the topic. Is there ever a time when you just have to use that complicated solution your teacher taught you? I use the simplest way to solve a problem to eliminate as many possibilities for error I can. Is this a bad habit?

Mathematicians are lazy people. They will try to find the easiest solution as possible and make sure they don't work hard. So no, it's not a bad habit
However, the simplest method might sometimes only work in special cases, while a more complicated method works more generally. So this is why you sometimes see people solving things the hard way. They just want to illustrate a method that always works.

For example, if you want to solve systems with $2$ equations and $2$ unknowns, then this is very simple using the simpler methods. However, there is also something called the matrix method (or Gauss-Jordan), which is a little bit more complicated and definitely overkill. However, for more complicated cases like $4$ equations and $4$ unkowns, the Gauss-Jordan method with matrices is definitely the easiest one. So people show you the Gauss-Jordan method in the simpler case just so you can get used to it.

As for your other questions. If you can find the answer easier by eliminating $x$, then go for it. In math there are no predetermined steps you are forced to take. You just need to make sure all your steps are logically correct. As long as that's true, you can do whatever you want to solve the problem! I don't know why your teacher eliminated $y$, you should ask him/her.

 

[Tyrion101]

Thank you, I've had both types of teachers, one would not accept any creativity in answering questions, the other would give extra credit, or not mark wrong for not doing it their way.

 

[micromass]

Thank you, I've had both types of teachers, one would not accept any creativity in answering questions

That's pretty sad. Well, of course it still needs to be correct. But if every step is logically correct, then it's pretty sad that the teacher won't give it full marks. Mathematics is not about following some predetermined scheme like computers. Mathematics is creativity!

 

[Tyrion101]

I think that's why I've hated math for so many years, and why I was confused when people had many different ways of solving an equation, or problem. And maybe I just misunderstood that they were only trying to illustrate something, but to me it felt like they were saying there is just one way to the answer. (Obviously not referring to problems involving just arithmetic, 2+2 never equals 5)

 

[micromass]

I think that's why I've hated math for so many years, and why I was confused when people had many different ways of solving an equation, or problem. And maybe I just misunderstood that they were only trying to illustrate something, but to me it felt like they were saying there is just one way to the answer. (Obviously not referring to problems involving just arithmetic, 2+2 never equals 5)

One of the nice things about math is that every question has a unique solution, but many ways to find the solution. However, showing more than one way is often confusing to students, so I guess the teachers like to follow some predetermined scheme to solve the problem. Sad though that it's contrary to the spirit of mathematics.

Obviously, Feynman always says it better than me:

https://www.youtube.com/watch?v=5ZED4gITL28