[ Algebra ] The Swapping Trick for Differences

What is - (a – b)?

Do you have a quick way to simplify this? It turns out that the answer is very easy! Just swap the two quantities. For example, -( 3 - 5 ) = 2. This is the same as result of 5 – 3. How do we know that -( 3 - 5 ) = 2? Well, we know that -( 3 - 5 ) is - (-2) and we saw yesterday that this is the same as 2.

In general, -(a - b) = b - a

Just swap the a and the b. How do we know this always works? The proof is very easy: -(a - b) = -a + b = b - a. Here is an example of using the swapping trick to solve certain algebraic fraction equation.

We can quickly make the equation easier by horizontally swapping the 1 and 2x and changing the attached + to -. This equalises the denominators. Now we can just subtract the numerators to get 1. But x still appears below as part of the denominator. What do we do? Just reciprocate (vertically flip) both sides of the equation. We easily get 2x = 4 i.e. x = 2.