How wide is the river?

[Alan Anderson]

Originally Posted by sanddrag

With the equation method of solving, how is it valid to divide the two equations? I know you are trying to eliminate the speed variable but I have never heard of that method (dividing equations) of solving before. Can someone please explain why that works?

Why it works is easy.

Consider that an equation means that the values on each side of the equals sign are, well, equal. It's perfectly valid to do the same arithmetic operation to each side of an equation, e.g. add 5, or multiply by 12, or subtract x, or divide by a.

Start with the equation a / u = (w-a) / v. You can accept multiplying both sides by u, right? You can also divide both sides by something more complicated, such as (w+b) / u. Here's the fun part: if you know that (w+b) / u = (2w-b) / v, you can divide the left side of the original equation by the left side of the new equation, while dividing the other side of the original equation by the other side of the new one. You're still dividing both sides by the same thing.