Fill in the blanks using commutatvity
of multiplication 3x(-4) =
X 3
Answers
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.
Why is the following true? 2(x + y) = 2x + 2y
Since they distributed through the parentheses, this is true by the Distributive Property.
Use the Distributive Property to rearrange: 4x – 8
algebra
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.
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
Simplify 2(3x), and justify your steps.
In this case, they do want me to simplify, but I have to say why it's okay to do... just exactly what I've always done. Here's how this works:
Why is it true that 2(3x) = (2×3)x?
Since all they did was regroup things, this is true by the Associative Property.
following:
4 × 3 × x
4 × x × 3
3 × x × 4
x × 3 × 4
x × 4 × 3
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
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!
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
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
Why is it true that 3(4 + x) = 3(x + 4)?
All they did was move stuff around.
Commutative Property
Why is 3(4x) = (3×4)x?
All they did was regroup.
Associative Property
Why is 12 – 3x = 3(4 – x)?
They factored.