Yesterday I named a property and claimed that it’s useful. Today I show how to put it to use.

In yesterday’s post I showed what I call the **Opposite Differences Property**, a property that** **tells us that the answer we get when we subtract one number from another, a – b, is the numerical opposite of what we get when we switch the order of the numbers, b – a.

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In other words, this property assures us that a – b = – (b – a)

A quick example of this with actual numbers: 7 – 4 = – (4 – 7), since 7 – 4 = 3, and 4 – 7 = – 3.

The property tells us that the two results are mathematical opposites of each other: 3 = – (– 3)

This is handy to know in general, but it’s especially useful when you’re simplifying complex algebraic fractions.

First, consider a simple algebraic fraction: (x – y) / (y – x)

The **Opposite Differences Property** tells us that the numerator’s difference is the opposite of the denominator’s difference. That means that this fraction simplifies to – 1.

Not convinced? Try plugging in a couple of numbers: let x = 9 and let y = 2.

The fraction then gives us: (9 – 2) / (2 – 9), and that simplifies to 7 / – 7. But what is 7/– 7? It’s just – 1.

This holds true no matter what numbers we use; that’s what we mean by calling this a “property.” It works in general.

So we can confidently assert that, in general: (x – y) / (y – x) = – 1

[To be mathematically proper, I must add one caveat. This property holds in all situations **but one**. The variable x cannot equal y, for then we would have 0 / 0, which is “indeterminate.” That’s a fancy term, but it means that it’s value can change depending on the situation. Stuff we don’t need to worry about now.]

But back to the power of the property … suppose we want to simplify the algebraic fraction:

(x + y) (x – y) / (y + x) (y – x)

Looks complicated, right? But we have the power to wrestle this to the ground.

First, using the Commutative Property, we know that the numerator’s (x + y) term = the denominator’s (y + x) term, so those two quantities simplify to 1/1.

And also, using our newly coined **Opposite Differences Property**, we see that the fraction made up of the other terms:

(x – y) / (y – x) = – 1

So (x + y)(x – y) / (y + x)(y – x) simplifies to (1) (– 1) = – 1

As another example, suppose you want to simplify this fraction:

abc (ab – b) / bc (b – ab)

First, the b and c in the numerator cancel with the b and c in the denominator. So as far as these plain variables are concerned, we’re left with only an ‘a’ in the numerator.

Secondly, the numerator’s (ab – b) is the **opposite difference** of the denominator’s (b – ab), so that part of the fraction turns into – 1.

Putting the pieces together, we have ‘a’ times – 1, which simplifies to just: – a

So that, – a, is the beautiful, simplified version of that big, ugly algebraic fraction.

As you do more and more algebraic fraction problems, you’ll find that the **Opposite Differences Property** comes in handy time and again. Learn it, use it, use it again.

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