Matching algebraic properties to their definitions on a worksheet.

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 the order of addition doesn’t matter:
> $ a + b = b + a $

→ (H) $ a + b = b + a $ ✔

✔ Match: (3) → (H)

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(4) Multiplicative Identity
The multiplicative identity is 1, because multiplying any number by 1 gives the same number:
> $ 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
This says grouping doesn't affect 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 affect 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 $ a = b $, then adding the same value to both sides keeps 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 to a number leaves it unchanged:
> $ x + 0 = x $

→ (F) $ x + 0 = x $ ✔

✔ Match: (11) → (F)

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Now check if all letters are used only once:

Used:
- (A) – reflexive
- (B) – multiplicative equality
- (C) – commutative mult.
- (D) – additive equality
- (E) – distributive
- (F) – additive identity
- (G) – multiplicative identity
- (H) – commutative add.
- (J) – transitive
- (M) – associative mult.
- (N) – associative add.

Leftover: (I), (K), (L)

But we have 11 items on the left and 14 on the right. So not all options are used — that’s fine.

Let’s verify if any were missed.

Wait — did we use (I), (K), (L)? No. But we have only 11 properties, so only 11 matches needed.

So we’re good.

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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

✔ All matched correctly.

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