Associative, Commutative, and Distributive Properties
Purplemath
There are three basic properties of numbers, and your textbook will probably have just a little section on these properties, somewhere near the beginning of the course, and then you'll probably never see them again (until the beginning of the next course). My impression is that covering these properties is a holdover from the "New Math" fiasco of the 1960s. While the topic will start to become relevant in matrix algebra and calculus (and become amazingly important in advanced math, a couple years after calculus), they really don't matter a whole lot now.
Why not? Because every math system you've ever worked with has obeyed these properties! You have never dealt with a system where a×b did not in fact equal b×a, for instance, or where (a×b)×c did not equal a×(b×c). Which is why the properties probably seem somewhat pointless to you. Don't worry about their "relevance" for now; just make sure you can keep the properties straight so you can pass the next test. The lesson below explains how I keep track of the properties.
Distributive Property
The Distributive Property is easy to remember, if you recall that "multiplication distributes over addition". Formally, they write this property as "a(b + c) = ab + ac". In numbers, this means, for example, that 2(3 + 4) = 2×3 + 2×4. Any time they refer in a problem to using the Distributive Property, they want you to take something through the parentheses (or factor something out); any time a computation depends on multiplying through a parentheses (or factoring something out), they want you to say that the computation used the Distributive Property.
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Why is the following true? 2(x + y) = 2x + 2y
Since they distributed through the parentheses, this is true by the Distributive Property.
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Use the Distributive Property to rearrange: 4x – 8
The Distributive Property either takes something through a parentheses or else factors something out. Since there aren't any parentheses to go into, you must need to factor out of. Then the answer is:
By the Distributive Property, 4x – 8 = 4(x – 2).
"But wait!" I hear you cry; "the Distributive Property says multiplication distributes over addition, not over subtraction! What gives?" You make a good point. This is one of those times when it's best to be flexible. You can either view the contents of the parentheses as the subtraction of a positive number ("x – 2") or else as the addition of a negative number ("x + (–2)"). In the latter case, it's easy to see that the Distributive Property applies, because you're still adding; you're just adding a negative.
The other two properties come in two versions each: one for addition and the other for multiplication. (Yes, the Distributive Property refers to both addition and multiplication, too, but it refers to both of the operations within just the one rule.)
Associative Property
The word "associative" comes from "associate" or "group"; the Associative Property is the rule that refers to grouping. For addition, the rule is "a + (b + c) = (a + b) + c"; in numbers, this means 2 + (3 + 4) = (2 + 3) + 4. For multiplication, the rule is "a(bc) = (ab)c"; in numbers, this means 2(3×4) = (2×3)4. Any time they refer to the Associative Property, they want you to regroup things; any time a computation depends on things being regrouped, they want you to say that the computation uses the Associative Property.
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Rearrange, using the Associative Property: 2(3x)
They want me to regroup things, not simplify things. In other words, they do not want me to say "6x". They want to see me do the following regrouping:
(2×3)x
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Simplify 2(3x), and justify your steps.
In this case, they do want me to simplify, but I have to way why it's okay to do... just exactly what I've always done. Here's how this works:
2(3x) : original (given) statement
(2×3) x : by the Associative Property
6x : simplification of 2×3
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Why is it true that 2(3x) = (2×3)x?
Since all they did was regroup things, this is true by the Associative Property.
Commutative Property
The word "commutative" comes from "commute" or "move around", so the Commutative Property is the one that refers to moving stuff around. For addition, the rule is "a + b = b + a"; in numbers, this means 2 + 3 = 3 + 2. For multiplication, the rule is "ab = ba"; in numbers, this means 2×3 = 3×2. Any time they refer to the Commutative Property, they want you to move stuff around; any time a computation depends on moving stuff around, they want you to say that the computation uses the Commutative Property.
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Use the Commutative Property to restate "3×4×x" in at least two ways.
They want me to move stuff around, not simplify. In other words, my answer should not be "12x"; the answer instead can be any two of the following:
4 × 3 × x
4 × x × 3
3 × x × 4
x × 3 × 4
x × 4 × 3
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Why is it true that 3(4x) = (4x)(3)?
Since all they did was move stuff around (they didn't regroup), this statement is true by the Commutative Property.
Worked examples
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Simplify 3a – 5b + 7a. Justify your steps.
I'm going to do the exact same algebra I've always done, but now I have to give the name of the property that says its okay for me to take each step. The answer looks like this:
3a – 5b + 7a : original (given) statement
3a + 7a – 5b : Commutative Property
(3a + 7a) – 5b : Associative Property
a(3+7) – 5b : Distributive Property
a(10) – 5b : simplification (3 + 7 = 10)
10a – 5b : Commutative Property
The only fiddly part was moving the "– 5b" from the middle of the expression (in the first line of my working above) to the end of the expression (in the second line). If you need help keeping your negatives straight, convert the "– 5b" to "+ (–5b)". Just don't lose that minus sign!
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Simplify 23 + 5x + 7y – x – y – 27. Justify your steps.
I'll do the exact same steps I've always done. The only difference now is that I'll be writing down the reasons for each step.
23 + 5x + 7y – x – y – 27 : original (given) statement
23 – 27 + 5x – x + 7y – y : Commutative Property
(23 – 27) + (5x – x) + (7y – y) : Associative Property
(–4) + (5x – x) + (7y – y) : simplification (23 – 27 = –4)
(–4) + x(5 – 1) + y(7 – 1) : Distributive Property
–4 + x(4) + y(6) : simplification (5 – 1 = 4, 7 – 1 = 6)
–4 + 4x + 6y : Commutative Property
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Simplify 3(x + 2) – 4x. Justify your steps.
3(x + 2) – 4x : original (given) statement
3x + 3×2 – 4x : Distributive Property
3x + 6 – 4x : simplification (3×2 = 6)
3x – 4x + 6 : Commutative Property
(3x – 4x) + 6 : Associative Property
x(3 – 4) + 6 : Distributive Property
x(–1) + 6 : simplification (3 – 4 = –1)
–x + 6 : Commutative Property
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Why is it true that 3(4 + x) = 3(x + 4)?
All they did was move stuff around.
Commutative Property
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Why is 3(4x) = (3×4)x?
All they did was regroup.
Associative Property
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Why is 12 – 3x = 3(4 – x)?
They factored.
Distributive Property
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