Solved period Batch 4fb17000 Algebraic Properties Version 1 ... - Free Printable

Let's solve this step by step. We are asked to match each algebraic property (on the left) with its correct definition (on the right). The definitions are labeled (A) through (N), and we need to match them correctly.

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Step 1: Understand each property

We’ll go through each numbered property and find its matching definition.

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(1) Distributive Property
This property states that multiplying a number by a sum is the same as multiplying it by each term and then adding:
> $ a(b + c) = ab + ac $

Look at the options:
→ (E) $ a(b + c) = ab + ac $ ✔

✔ Match: (1) → (E)

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(2) Reflexive Property
This says that any quantity is equal to itself:
> $ a = a $

→ (A) $ a = a $ ✔

✔ Match: (2) → (A)

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(3) Commutative Property of Addition
This means order doesn't matter in addition:
> $ a + b = b + a $

→ (H) $ a + b = b + a $ ✔

✔ Match: (3) → (H)

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(4) Multiplicative Identity
This means multiplying by 1 leaves a number unchanged:
> $ 1x = x $

→ (G) $ 1x = x $ ✔

✔ Match: (4) → (G)

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(5) Multiplicative Property of Equality
If two quantities are equal, multiplying both sides by the same number keeps them equal:
> If $ a = b $, then $ ax = bx $

→ (B) $ \text{if } a = b \text{ then } ax = bx $ ✔

✔ Match: (5) → (B)

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(6) Associative Property of Multiplication
Grouping doesn’t matter in multiplication:
> $ (ab)c = a(bc) $

→ (M) $ (ab)c = a(bc) $ ✔

✔ Match: (6) → (M)

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(7) Associative Property of Addition
Grouping doesn’t matter in addition:
> $ (a + b) + c = a + (b + c) $

→ (N) $ (a + b) + c = a + (b + c) $ ✔

✔ Match: (7) → (N)

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(8) Commutative Property of Multiplication
Order doesn’t matter in multiplication:
> $ ab = ba $

→ (C) $ ab = ba $ ✔

✔ Match: (8) → (C)

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(9) Additive Property of Equality
If two things are equal, adding the same value to both sides maintains equality:
> If $ a = b $, then $ a + x = b + x $

→ (D) $ \text{if } a = b \text{ then } a + x = b + x $ ✔

✔ Match: (9) → (D)

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(10) Transitive Property of Equality
If $ a = b $ and $ b = c $, then $ a = c $

→ (J) $ \text{if } a = b \text{ and } b = c \text{ then } a = c $ ✔

✔ Match: (10) → (J)

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(11) Additive Identity
Adding zero doesn’t change a number:
> $ x + 0 = x $

→ (F) $ x + 0 = x $ ✔

✔ Match: (11) → (F)

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Now let’s check if all letters are used only once:

Used:
- (A) – reflexive
- (B) – multiplicative property of equality
- (C) – commutative of multiplication
- (D) – additive property of equality
- (E) – distributive
- (F) – additive identity
- (G) – multiplicative identity
- (H) – commutative of addition
- (J) – transitive
- (M) – associative of multiplication
- (N) – associative of addition

Left unused:
- (I) $ (-a)(-b) = ab $ → This is about multiplying negatives; not one of the listed properties
- (K) $ (ab)^n = a^n b^n $ → Power of a product
- (L) $ (-a)b = a(-b) = -ab $ → Negative multiplication property

But none of these are named in the list, so they are extra.

So our matches are correct.

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✔ Final Answers:

| Number | Property | Match |
|--------|-----------------------------------------|-------|
| (1) | Distributive property | E |
| (2) | Reflexive property | A |
| (3) | Commutative property of addition | H |
| (4) | Multiplicative identity | G |
| (5) | Multiplicative property of equality | B |
| (6) | Associative property of multiplication | M |
| (7) | Associative property of addition | N |
| (8) | Commutative property of multiplication | C |
| (9) | Additive property of equality | D |
| (10) | Transitive property of equality | J |
| (11) | Additive identity | F |

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✔ Answer Key:

(1) E
(2) A
(3) H
(4) G
(5) B
(6) M
(7) N
(8) C
(9) D
(10) J
(11) F

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