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Simplification

Addition and subtraction


Parts of a mathematical problem can be added or subtracted from each other, if they are of the same kind, that is, if they are all numbers or if they are the same combination of letters and exponents.

Here are four examples:

a+2a=3a
2a-a=a
2b+2a+b=3b+2a
2b^2+3ab-4a+b^2=3b^2+3ab-4a

In the last example, we end up with three different parts:

b^2 and ab and a

which cannot be simplified any further.

The Commutative Property of Addition

When simplifying an expression it’s a good thing to keep the commutative property of addition in mind, saying that the order of the addends doesn’t matter.

a + b = b + a

The commutative property doesn’t apply to subtraction, because you cannot change the order of the minuend and the subtrahend without changing the result. But it is possible, however, to transform a subtraction problem into an addition problem, if you consider the last number as a negative number that has to be added to the first number, instead of two positive number being subtracted from each other.

In order to show this we write a minus sign in front of the last number and then put it into parentheses.

a - b = a + (-b)

Now we have an addition problem instead, where it’s possible to change the order of the parts.

a + (-b) = (-b) + a = -b + a

Multiplication and division


According to the commutative property of multiplication you can change the order of the factors without changing the product.

a*b=b*a

The commutative property doesn’t apply to division, since you cannot change the order of the divisor and the dividend without changing the result.