Dear all, I am confused by the following concept contained in the image attached. The equation boxed in red seems like a much more efficient method for simplifying, but I can not find a reason why this is true! My clumsy method is boxed in green and is nothing like that boxed in red. They both come to the same answer, but I can not see how the first method makes algebraic sense.
I don't know that I would consider the result to be "simplified", since instead of having one voltage being equal to a constant times another voltage, you have a difference between two voltages being equal to a constant times one of the voltages involved in the difference. Which seems simpler to you? But if, for some reason, you wanted an expression for (V2-V1) and didn't mind that it might involve V1 and or V2 on the right hand side, then the way you did it in green is the most obvious way to approach it. It clearly reduces to the result in red by factoring out V2 from the numerator. I don't know why you are keeping the parentheses around the R even though seem to recognize that none is needed around the S. You just need to recognize that V2[(R)-S] is the same as (R-S)V2.
They didn't include a couple of steps. My experience has been that people who are good at Maths have a habit of doing part on the work in their head,& jumping steps in what is written down. It is an efficient way of working,but not of teaching. Your answer needs to be simplified a bit further,but you got it OK.
YES! One of the worst habits of people teaching the basics. Making things to look like something popping up from the hat is bad for the student trying to understand / learn a process. From what I have seen, is quite common in text books where the degree of detail vary a lot for different examples.
But, giving the devil his due, you have to assume some minimum level of proficiency from the students, particularly regarding being able to perform the mechanics of mathematical manipulations. After all, I think most people would agree that it would be just as bad for a calculus professor to always be explicitly be carrying out the distribution of A(B+C) saying that it is AB+AC. Similarly, when examples involve new concepts, the detail with which those examples are worked out should be pretty deep. But as those concepts are supposed to be becoming well-learned, it should be sufficient to expect students to be able to recognize where they need to be applied and to do so with little or no explicit prompting. After all, when they start working "real" problems, they aren't going to have someone or some book showing them how and what to do step by step. In this case, we don't know what level this course is at. So I can't say whether the amount of detail provided was insufficient or not. Frankly, I don't think any further detail was warranted for any course for which Algebra I is a pre-req.