Laws of Algebra

Correct Answer: A
The commutative law states that you can swap the order of numbers in addition or multiplication and get the same result. Option A shows exactly this—6 + 2 equals 2 + 6, proving order doesn't matter.

Why the others are wrong:
- B demonstrates the *associative* law (changing which numbers you group together, not their order)
- C shows the *distributive* law (multiplying across a sum)
- D also shows the *associative* law for multiplication

Correct Answer: A
The Associative Law says you can regroup numbers being multiplied without changing the result. Option A shows this perfectly: moving the parentheses from (2 × 3) to (4 × 2) doesn't change the answer—both equal 24.

Why the others are wrong:
- B shows the Commutative Law (order doesn't matter), not associative
- C shows the Distributive Law (multiplication distributed over addition)
- D shows the Commutative Law of Addition, not multiplication at all

Correct Answer: A. 5a + 5b

The distributive law says you multiply the number outside the parentheses by *each* term inside. Here, 5 multiplies both *a* and *b*, giving you 5·a + 5·b = 5a + 5b.

Why the others are wrong:
- B (a + b + 5): This just adds 5 to the expression instead of multiplying everything by 5.
- C (a + 5b): You only multiplied 5 by *b*, forgetting to multiply 5 by *a*.
- D (5a + b): You only multiplied 5 by *a*, forgetting to multiply 5 by *b*.

# Correct Answer: A

Why A is correct: The commutative law of multiplication states that you can multiply numbers in any order and get the same result. This equation shows exactly that—flipping the order of 7 and 4 gives the same product.

Why the others are wrong:
- B: This demonstrates the *associative* law of addition (grouping doesn't matter), not multiplication
- C: This is the *distributive* law (multiplying across a sum)
- D: This is the *associative* law of multiplication (grouping factors differently), not the commutative law

Why A is correct:
The distributive law of division states that (a + b)/c = a/c + b/c. You divide each term in the numerator by the denominator separately, so (x + y)/3 becomes x/3 + y/3.

Why the others are wrong:
- B: (x + y)/6 divides by 6 instead of 3—this doesn't follow the distributive property.
- C: (x)/(y) + 3 incorrectly divides x by y and adds 3, which has nothing to do with distributing division over addition.
- D: (3)/(x) + (3)/(y) flips the fraction upside down (reciprocal), which is wrong—we're not dividing by (x + y).

A is correct. The Associative Law of Addition states that how you *group* numbers with parentheses doesn't change the sum: (2 + 4) + 5 = 2 + (4 + 5) = 11 either way.

B is wrong — that's the Commutative Law of Multiplication (order doesn't matter).

C is wrong — that's the Distributive Law (multiplying across a sum).

D is wrong — that's the Commutative Law of Addition (order doesn't matter), not about grouping.

A. 3(a - 2b) ✓

Distribute the 3: 3(a) - 3(2b) = 3a - 6b. This matches exactly.

B. 3a + 6b ✗
This has a plus sign instead of minus, so it's the wrong operation.

C. a(3 + 2b) ✗
Distributing gives 3a + 2ab, which has an extra term (2ab) we don't want.

D. a - 2b - 2a - 4b ✗
Combining like terms: (a - 2a) + (-2b - 4b) = -a - 6b, which is different from our target.

A is correct because the associative law says you can regroup numbers being multiplied without changing the result. Here, the parentheses move from (2 × 5) to (5 × 4), but both sides equal 40—that's the associative property in action.

B is wrong because that's the commutative law of addition (the order changes, not the grouping).

C is wrong because that's the distributive property (multiplication distributed over addition).

D is wrong because that's the commutative law of multiplication (the order changes, but grouping stays the same).

# Explanation

A is correct: The Commutative Law of Multiplication states that the order of factors doesn't change the product. 8 · 3 = 3 · 8 perfectly demonstrates this—both equal 24.

B is wrong: This shows the Commutative Law of *Addition*, not multiplication (both use addition signs).

C is wrong: This is the *Distributive* Law, which shows how multiplication distributes over addition.

D is wrong: This demonstrates the *Associative* Law of Multiplication, showing that how you group factors doesn't change the product.

A. 4(2x + 3y) ✓

The greatest common factor (GCF) of 8x and 12y is 4. When you factor it out, you get 4(2x + 3y). You can check this by distributing: 4 × 2x = 8x and 4 × 3y = 12y. ✓

B. 3(4x + 6y) — While this expression equals 12x + 18y when distributed, not 8x + 12y. Also, 3 is not the GCF of 8 and 12.

C. 8(x + y.5) — This equals 8x + 12y, but it's not the fully factored form because you can still factor out more (the 4). Also, using decimals is awkward here.

D. 4x + 6y — This isn't factored at all; it's still in expanded form and doesn't equal the original expression 8x + 12y.

A is correct: When you distribute -3, you multiply it by each term: -3 times x gives -3x, and -3 times -4 gives +12 (negative times negative = positive). So -3(x - 4) = -3x + 12.

B is wrong: This forgets that -3 × -4 = +12, not -12.

C is wrong: The first term should be negative (-3x), not positive.

D is wrong: Both signs are wrong—the first term should be negative and the second should be positive.

Why A (9) is correct:
- Left side: a + (b + c) = 2 + (3 + 4) = 2 + 7 = 9
- Right side: (a + b) + c = (2 + 3) + 4 = 5 + 4 = 9
- Both sides equal 9, proving the associative law. The common value is 9.

Why the others are wrong:
- B (6): This is just a + b, not the full expression.
- C (12): This is a + b + c but calculated incorrectly (2 + 3 + 4 = 9, not 12).
- D (5): This is only a + b, not the complete answer.

Why A is correct:
Distribute -2 to each term inside the parentheses:
- -2 × x = -2x
- -2 × y = -2y
- -2 × (-z) = +2z (negative times negative = positive)

Result: -2x - 2y + 2z

Why the others are wrong:
- B: Incorrectly distributes -2 to y as +2y instead of -2y
- C: Changes all signs to positive, as if distributing +2 instead of -2
- D: Forgets that -2 times -z should be positive, not negative

Why A is correct:
The distributive law of division means dividing each term in the numerator by the denominator: (3x)/3 + 6/3 = x + 2.

Why the others are wrong:
- B: Only divides the first term by 3, leaving the second term unchanged—doesn't apply distribution.
- C: Doesn't divide anything by 3; it just removes the parentheses incorrectly.
- D: Inverts the operation (puts 3 and x in denominators) rather than dividing the original numerator by 3.

• Left side: x(y + z) = 1(2 + 3) = 1(5) = 5
• Right side: xy + xz = (1)(2) + (1)(3) = 2 + 3 = 5
• Both sides equal 5, so the answer is B. 5 ✓

Why the others are wrong:
- A (6): Wrong—this might come from adding all three variables, but that's not what the equation asks for
- C (7): Wrong—this might come from misadding or including extra terms
- D (3): Wrong—this is just the value of z, not the result of either side of the equation

Correct Answer: A (-2x + 5)

Notice both terms share the common factor (2x - 5), so you can factor it out: (-4 + 3)(2x - 5) = -1(2x - 5) = -2x + 5.

Why the others are wrong:

• B (2x - 5): This is just the common factor itself, not the result of factoring it out.
• C (-6x + 10): This comes from incorrectly distributing without factoring; it doesn't simplify the expression.
• D (-x + 15): This results from arithmetic errors in combining the terms.

Why A is correct:
The associative law lets you change which terms are grouped together using parentheses. Option A regroups by moving the parentheses from around "(x + 1)" to around "(3 + x + 1)", which is a valid regrouping.

Why the others are wrong:
- B: This removes the parentheses entirely, so it's not regrouped—it's just written differently.
- C: While this is also a valid regrouping mathematically, it's not the answer the question is asking for.
- D: This changes the *order* of terms (commutative law), not the grouping (associative law).

# Why A is Correct

A correctly applies the distributive law formula given in the question. When you substitute x = 2 and y = 1 into x(y + 3) = xy + 3x, you get exactly 2(1 + 3) = 2(1) + 2(3). Both sides equal 8, confirming the law works.

# Why Others Are Wrong

B: Missing the multiplication—it just shows 2(1 + 3) = 1 + 3, which would mean multiplying by 2 does nothing.

C: Incorrectly drops the parentheses but only adds the numbers (1 + 3), ignoring the multiplication by 2.

D: Leaves out the multiplication entirely and just adds all three numbers together, which doesn't represent the distributive property.

A is correct because the Distributive Law allows you to "factor out" a common term from an addition. Here, *a* is common to both *ab* and *ac*, so you pull it out front: ab + ac = a(b + c). This is factoring in action.

B is wrong — that's the Commutative Law of Addition (order doesn't matter).

C is wrong — that's the Associative Law of Addition (grouping doesn't matter).

D is wrong — that's the Commutative Law of Multiplication (order doesn't matter).

Correct Answer: 9

Substitute a = 3 and b = -2 into a(b + a) - ab:
- a(b + a) - ab = 3(-2 + 3) - 3(-2) = 3(1) - (-6) = 3 + 6 = 9 ✓

Why the others are wrong:

• B (6): You'd get this if you forgot to add back the -ab term, stopping at just a(b + a) = 3.
• C (3): You'd get this if you made an error combining like terms or miscalculated the final addition.
• D (12): You'd get this if you incorrectly distributed or added the terms—for example, treating -ab as positive.

A is correct because it shows the distributive law applied incorrectly. The 3 is only distributed to the 4, not to the 2—it should be $3 × 4 + 3 × 2$.

B is correct application of the distributive law: you multiply 3 by each term inside the parentheses.

C is correct because it simplifies B further ($12 + 6 = 18$), showing the distributive law was applied properly.

D is correct because it's the final simplified answer ($18$), which results from correctly applying the distributive law.

• Correct answer (A: 10): The associative law says you can regroup numbers when adding without changing the result. So (2 + 5) + 3 = 2 + (5 + 3) = 2 + 8 = 10. Either way, the answer is 10.

• B (9): This would be 2 + 5 + 2, which is incorrect.

• C (8): This is just 5 + 3, missing the 2 entirely.

• D (11): This incorrectly adds an extra number or miscalculates the sum.

Correct answer: A. 4 × (5 × 2)

The associative law lets you regroup numbers in multiplication without changing the result. The original expression groups the first two numbers: (4 × 5) × 2. Option A regroups it as 4 × (5 × 2)—same numbers, same operation, just moved the parentheses.

Why the others are wrong:
- B: Changes × to +, which violates the associative law (it's a different operation)
- C: Removes parentheses and adds instead of multiplies—completely changes the problem
- D: Mixes addition and multiplication, not a valid regrouping of the original expression

A. 10, Associative Law ✓

The expression equals 10 (5 + 5 = 10), and the Associative Law allows us to regroup addends without changing the result: (3 + 2) + (4 + 1) = 3 + 2 + 4 + 1 = 10.

Why the others are wrong:
- B (Distributive Law): This law involves multiplying across groups (like 2(3 + 4)), not just regrouping addition.
- C (Commutative Law): This law lets us *reorder* terms (3 + 5 = 5 + 3), but we're *regrouping*, not reordering.
- D (9): The math is wrong; it still equals 10.

Correct answer: A

The commutative law of multiplication states that you can multiply numbers in any order and get the same result: $a × b = b × a$. Option A shows exactly this—switching the order of 6 and 7 doesn't change the product.

Why the others are wrong:
- B shows the *associative* law (changing which numbers you group), not commutative
- C shows the *distributive* law (multiplication distributed over addition)
- D is just restating the same thing; it doesn't demonstrate any multiplication law

Correct answer: A. $6x + 8y$

The distributive law says multiply the number outside the parentheses by each term inside: 2 × 3x = 6x, and 2 × 4y = 8y.

• B ($5x + 6y$): Wrong—you can't add 2 + 3 or 2 + 4; you must multiply.
• C ($3x + 8y$): Wrong—you multiplied 2 × 4y correctly but forgot to multiply 2 × 3x.
• D ($2x + 4y$): Wrong—this just rewrites the original expression without distributing anything.

A is correct because the commutative law does NOT apply to subtraction. $5 − 3 = 2$, but $3 − 5 = −2$—they're not equal, so this is incorrect.

B, C, and D are all correct applications:
- B: Addition is commutative ($7 + 2 = 2 + 7 = 9$) ✓
- C: Multiplication is commutative ($6 × 9 = 9 × 6 = 54$) ✓
- D: Addition is commutative ($12 + 3 = 3 + 12 = 15$) ✓

The key: The commutative law only works for addition and multiplication, not subtraction or division.

A is correct: The distributive law says you multiply each term in the first group by each term in the second group. So: x times (a + b) gives xa + xb, and y times (a + b) gives ya + yb. Combined: xa + xb + ya + yb.

B is wrong: This only multiplies the "first" terms and "last" terms—it skips the cross products (xb and ya).

C is wrong: This just adds all four variables together with no multiplication, which ignores what the distributive law requires.

D is wrong: This incorrectly combines a and b into a single product (ab) before multiplying, rather than distributing to each term separately.

Why A is correct:
The Identity Property of Addition states that adding 0 to any number gives you that same number back—0 is the "identity" because it doesn't change the value. That's exactly what 9 + 0 = 9 shows.

Why the others are wrong:
- B (Commutative Law): This property says you can swap numbers (like 3 + 5 = 5 + 3), but that's not what's special about adding 0 here.
- C (Distributive Law): This involves multiplying across a sum, like 2(3 + 4) = 2(3) + 2(4)—not relevant here.
- D (Associative Law): This property groups numbers differently (like (2 + 3) + 4 = 2 + (3 + 4)), but we only have two numbers.

Why A is correct:
The associative law says you can regroup numbers when adding without changing the result. (6 + 4) + (3 + 7) = 10 + 10 = 20.

Why the others are wrong:
- B (10): This is just one of the groups (6 + 4), not the full answer.
- C (18): This might come from miscalculating; perhaps adding 6 + 4 + 3 + 5 by mistake.
- D (16): This is incorrect arithmetic; there's no valid grouping that gives 16.

Why A is correct:
Distribute first: 5(x + 2) = 5x + 10 and -3(x - 1) = -3x + 3. Then combine: 5x + 10 - 3x + 3 = 2x + 13.

Why the others are wrong:
- B: This shows the distribution step but isn't simplified—you need to combine like terms to finish.
- C: This incorrectly combines terms that shouldn't be combined (5x and -3x are like terms, but you're missing the constants +10 and +3).
- D: Wrong arithmetic; doesn't properly combine the like terms or constants.

Correct Answer: A

The Identity Property of Multiplication states that any number multiplied by 1 equals that number (a · 1 = a). This is the exact definition matching the given equation.

Why the others are wrong:
- B (Associative Law): Groups factors differently like (a · b) · c = a · (b · c)—not about multiplying by 1
- C (Distributive Law): Distributes multiplication over addition like a(b + c) = ab + ac—doesn't apply here
- D (Commutative Law): Says order doesn't matter (a · b = b · a)—not the reason 1 keeps a unchanged

Option A is correct because it doesn't follow the distributive law. The right side (12 + 2) only multiplies the first term by 3, leaving the second term unchanged—that's not how distribution works.

Why the others are wrong:
- B correctly distributes: 3 multiplies both 4 and 2, giving 12 + 6 ✓
- C correctly distributes: 6 multiplies both x and y ✓
- D correctly distributes: 2 multiplies both 5 and 7, giving 10 + 14 ✓

The distributive law requires multiplying the outside number by every term inside the parentheses. Option A fails to do this.

A. $9 + y + x$ ✓

The commutative law says you can add numbers in any order and get the same result. The original expression $x + 9 + y$ can be rearranged to $9 + y + x$—we've just moved the terms around.

Why the others are wrong:
- B ($y + 9x$): This multiplies 9 and x together, which changes the value—commutative law doesn't let us do that.
- C ($xy + 9$): This multiplies x and y, which also changes what we're calculating.
- D ($x + y + 9$): While this is also correct mathematically, it's not the answer shown—it's just a different rearrangement.

A is correct.

The Distributive Law states that a × (b + c) = a × b + a × c. Here, 7 is "distributed" across the addition (2 + 3), giving 7 × 2 + 7 × 3 = 14 + 21 = 35. Both sides equal 35, confirming the law works.

Why the others are wrong:
- B: While 35 = 35 is true, this describes the Commutative Law (order doesn't matter in addition/multiplication), which isn't what's being shown here.
- C: The Associative Law groups terms differently, like (a + b) + c = a + (b + c). This problem doesn't rearrange groupings; it distributes a multiplier.
- D: The math is wrong—14 × 21 = 294, not 35. Also, multiplication isn't the operation being distributed here.

Why A is correct:
The associative law lets us regroup terms. Combine all the variables: (x + y + z). Then add the constants: 1 + 2 + 3 = 6. Result: x + y + z + 6.

Why the others are wrong:
- B (sum = 5): Incorrectly adds only two constants (1 + 2 + 3 ≠ 5).
- C (sum = 7): Adds one too many (1 + 2 + 3 = 6, not 7).
- D (sum = 3): Only counts one constant instead of all three.

A is correct: Combine like terms—add the x terms (2x + 5x = 7x) and add the y terms (4y + y = 5y) to get 7x + 5y.

B is wrong: This multiplies x and y together instead of adding like terms separately.

C is wrong: This incorrectly subtracts the x terms (5x − 2x) instead of adding them.

D is wrong: This adds the coefficients incorrectly (getting 6x instead of 7x, and 3y instead of 5y).

Why A is correct:
The distributive law means finding the common factor of all terms, then factoring it out. Both 10x and 15y share a common factor of 5. When you divide: 10x ÷ 5 = 2x and 15y ÷ 5 = 3y, giving you 5(2x + 3y). Check: 5(2x + 3y) = 10x + 15y ✓

Why the others are wrong:
- B: This removes the common factor but doesn't use the distributive law—it's just subtraction, not factorization.
- C: This is just part of the answer; it's missing the common factor of 5.
- D: While 10 is a factor of 10x, it's not a factor of 15y, so this doesn't work for both terms.

Correct answer: A. $6 + 6x$

• First, distribute the 2: $2(3 + x) = 6 + 2x$
• Then add the $4x$: $6 + 2x + 4x = 6 + 6x$ ✓

Why the others are wrong:
- B ($6 + x$): You'd get this if you forgot to distribute the 2 to the $x$, or if you didn't combine like terms properly.
- C ($10x + 3$): This swaps the constant and variable terms—the 6 and $6x$ are flipped.
- D ($6x + 4$): This incorrectly uses 4 instead of 6 as the constant term.

A is correct because it changes both the grouping AND the order of the numbers. The associative law only allows you to move parentheses—not rearrange the actual numbers. Here 4 and 3 switched positions, which violates the rule.

B is wrong (it's actually correct): parentheses moved from $(1+2)+3$ to $1+(2+3)$, but all numbers stayed in the same order.

C is wrong (it's actually correct): same idea with multiplication—only grouping changed, numbers stayed in order.

D is wrong (it's actually correct): parentheses moved, numbers stayed in order—this correctly demonstrates the associative law.

A is correct because it directly matches the pattern ab + ac = a(b + c). Here, a = 2, b = 4, and c = 6, so you have 2×4 + 2×6 on the left side and 2(4 + 6) on the right side—exactly the factoring pattern.

B is wrong because 4×2 + 6 doesn't have a common factor in both terms the way the pattern requires (the second term is just 6, not 2×6).

C is wrong because 2 + 4 + 6 uses addition throughout, not multiplication combined with addition; it doesn't show the factoring pattern.

D is wrong because it reverses the equation and uses addition (2 + 4 + 6) instead of multiplication (2×4 + 2×6), so it's not true: 2(4 + 6) = 20, but 2 + 4 + 6 = 12.

A is correct. The commutative law of addition says you can swap the order of numbers being added and get the same result: $a + b = b + a$. This equation shows exactly that—switching 6 and 2 gives the same sum.

B is wrong — this is the *associative* law, which is about grouping with parentheses, not swapping order.

C is wrong — this shows the *distributive* law (multiplying across a sum).

D is wrong — while it does show the commutative property, it's for *multiplication*, not addition.

A is correct: The distributive law means finding the common factor of both terms. Both 14x and 21y are divisible by 7 (the greatest common factor), so we factor it out: 7(2x + 3y). Check: 7 × 2x = 14x ✓ and 7 × 3y = 21y ✓

B is wrong: This isn't factorised—it's just a different expression with no common factor removed.

C is wrong: This removes the common factor but doesn't show the factorisation properly; the original terms aren't recovered.

D is wrong: While 14 is a factor of 14x, it's not a factor of 21y, so it's not a common factor of both terms.

Correct answer: $22$

• First, simplify the expression: 3(2x + 4) - x = 6x + 12 - x = 5x + 12
• Then substitute x = 2: 5(2) + 12 = 10 + 12 = 22 ✓

Why others are wrong:
• B ($12$): You'd get this if you forgot to add the 12 constant
• C ($10$): This is just 5x without adding the constant
• D ($16$): This results from distributing incorrectly or making an arithmetic error

Why A is correct (the false statement):
Let's check: (10 - 2) - 3 = 8 - 3 = 5, but 10 - (2 - 3) = 10 - (-1) = 11. These don't equal each other, so subtraction is NOT associative. This statement claims something false.

Why the others are true:
- B: (4 + 5) + 2 = 9 + 2 = 11, and 4 + (5 + 2) = 4 + 7 = 11 ✓ Addition is associative.
- C: (2×3)×4 = 6×4 = 24, and 2×(3×4) = 2×12 = 24 ✓ Multiplication is associative.
- D: 3 + 6 = 9 and 6 + 3 = 9 ✓ Addition is commutative (order doesn't matter).

A is correct: (x + 2)² means (x + 2)(x + 2). Using distributive law: x·x + x·2 + 2·x + 2·2 = x² + 2x + 2x + 4 = x² + 4x + 4.

Why others are wrong:
- B forgot to combine the middle terms (2x + 2x should equal 4x, not 2x)
- C made the same middle-term error and also got the constant wrong (should be 4, not 2)
- D missed both the middle terms and the constant entirely

A is correct because the right side shows $(3 + 4) × 4$, not $(3 + 4) × 2$. The commutative law lets you swap the order of factors, but you can't change what the factors actually are—this example changes 2 to 4, which violates the law.

B is wrong (it's correct use) because $2 × 3 = 3 × 2$ perfectly demonstrates the commutative law: swapping the order gives the same result (both equal 6).

C is wrong (it's correct use) because $a × b = b × a$ is the definition of the commutative law itself.

D is wrong (it's correct use) because $5 × 1 = 1 × 5$ correctly swaps the factors and both equal 5.

Correct: A. $a × (b × c)$

The associative law lets you move parentheses around without changing the result. Here, you're moving them from around $(a × b)$ to around $(b × c)$, so $(a × b) × c = a × (b × c)$.

Why the others are wrong:
- B: Changes multiplication to addition—that's a different operation entirely.
- C: Mixes multiplication and addition—again, a different operation.
- D: Has addition instead of multiplication—doesn't follow the associative law for multiplication.

A is correct. The distributive law says you multiply the number outside the brackets by each term inside: 3 × x = 3x, and 3 × (–4) = –12, giving you 3x – 12.

B is wrong because you'd get a plus sign instead of minus—that would only happen if the original was 3(x + 4).

C is wrong because you forgot to multiply the x by 3; you only distributed to the –4.

D is wrong because you only multiplied the 4 by 3, not the x.

Why A is correct:
The greatest common factor (GCF) of 9a and 12b is 3. Factoring it out gives 3(3a + 4b), and you can verify: 3 × 3a + 3 × 4b = 9a + 12b ✓

Why the others are wrong:
- B: 9 doesn't divide evenly into 12b, so it's not a common factor
- C: While this technically works mathematically, 6 isn't the GCF—we should always use the *greatest* common factor
- D: Same issue—12 isn't the GCF, and it's not even a common factor of both terms

Why A is correct:
The associative law for multiplication states that how you group factors doesn't change the product: $(a × b) × c = a × (b × c)$. Option A shows exactly this—both sides equal 24, just with different grouping.

Why the others are wrong:
- B violates the associative law by mixing operations (multiplication and addition), which breaks the rule
- C is just a calculation, not demonstrating any law
- D shows the commutative law (order doesn't matter: $2 × 4 = 4 × 2$), not the associative law, since the grouping of factors changes but one factor position also swaps

Correct answer: A. 4x + 8

Distribute each term:
- 5(2x + 1) = 10x + 5
- -3(2x - 1) = -6x + 3

Combine: 10x + 5 - 6x + 3 = 4x + 8 ✓

Why others are wrong:
- B (2x + 2): Forgot to distribute the negative sign properly; would lose terms.
- C (10x - 6): Didn't combine like terms; just took the first distribution and ignored the second part.
- D (8x + 4): Error in distribution or combining—likely miscalculated the constants (5 + 3 ≠ 4).

Correct Answer: A
This uses both laws: the commutative law changes the *order* of numbers (3 + 4 becomes 4 + 3), AND the associative law changes which numbers are grouped together (the parentheses move from (3+4)+5 to 5+(4+3)).

Why the others are wrong:
- B: Only uses associative law (parentheses move, but the order 3, 4, 5 stays the same)
- C: Only uses commutative law (order changes, but there's no grouping/parentheses to show associative property)
- D: Only uses associative law (parentheses move, but order 3, 5, 4 stays the same)

Why A is correct:
This identity is *incorrect* (which is what the question asks for). Let's check: $(4 + 5) + 2 = 9 + 2 = 11$, and $4 + (2 + 5) = 4 + 7 = 11$. They're actually equal, so the "≠" symbol makes the statement false.

Why B is wrong:
This demonstrates the associative property of addition: $4 + (5 + 2) = 4 + 7 = 11$ and $(4 + 5) + 2 = 9 + 2 = 11$. It's a correct identity.

Why C is wrong:
This is the commutative property of multiplication: $6 × 3 = 18$ and $3 × 6 = 18$. It's correct.

Why D is wrong:
This is the distributive property: $2 × (3 + 1) = 2 × 4 = 8$ and $2×3 + 2×1 = 6 + 2 = 8$. It's correct.

Correct answer: A

This uses the distributive property: when a number multiplies two groups, you can factor it out. Since 25 multiplies both 4 and 6, you can write it as 25 × (4 + 6) = 25 × 10 = 250—much easier than calculating 100 + 150 separately.

Why the others are wrong:
- B ($25 × 4 × 6$): This gives 600, not the same result—you'd be multiplying instead of adding the 4 and 6.
- C ($(25 + 4) × 6$): This changes which numbers combine; it equals 174, not 250.
- D ($25 + (4 × 6)$): This ignores the distributive property entirely; it gives 49, which is completely different.

Why A ($12x$) is correct:
Expand both squares: $(x+3)^2 = x^2 + 6x + 9$ and $(x-3)^2 = x^2 - 6x + 9$. When you subtract them, the $x^2$ terms cancel and the $9$s cancel, leaving $6x - (-6x) = 12x$.

Why the others are wrong:
- B ($6x$): You'd get this if you only subtracted one of the $6x$ terms instead of combining both.
- C ($0$): This assumes all terms cancel, but the middle terms ($+6x$ and $-6x$) actually add together when subtracting.
- D ($9x$): This doesn't come from correctly expanding and simplifying the expression.

A is correct: The distributive law says multiply the number outside the parentheses (2) by *each* term inside. So: 2·x + 2·y + 2·z = 2x + 2y + 2z.

B is wrong: This adds 2 as a separate term instead of multiplying it by each term in the parentheses.

C is wrong: This only multiplies 2 into part of the expression (xy), leaving z unchanged—you must distribute to all terms.

D is wrong: This multiplies 2 only by x, but forgets to multiply 2 by y and z.

A is correct: The associative law of addition states that how you *group* numbers with parentheses doesn't change the sum: $(a + b) + c = a + (b + c)$. Option A shows exactly this—the same three numbers (2, 3, 4) added in the same order, just grouped differently.

B is wrong: This shows the *commutative* law (order doesn't matter), not the associative law (grouping doesn't matter).

C is wrong: This mixes addition and multiplication, and the equation is actually false ($10 ≠ 14$). The associative law applies to one operation at a time.

D is wrong: While both sides equal 9, this actually changes the *order* of the numbers being grouped, so it demonstrates both commutative and associative properties together—not the associative law alone.

# Explanation

Why A is correct:
(7 + 2) + 5 = 9 + 5 = 14. The associative law lets you regroup which numbers you add together first—7 + (2 + 5) = 7 + 7 = 14. Both give the same answer, showing the law works.

Why B is wrong:
This just swaps the order of 7 and 2, using the commutative law, not the associative law. The associative law changes which numbers are grouped together with parentheses, not the order they appear.

Why C is wrong:
The answer is 13, not 14, so the math is wrong. Also, it introduces a 6 that wasn't in the original problem.

Why D is wrong:
While it correctly equals 14, it moves 5 to the front. This changes the *order* of numbers (commutative law), not just their *grouping* (associative law).

A is correct: The associative law of multiplication says you can regroup factors with parentheses and get the same answer: $(a × b) × c = a × (b × c)$. Option A shows exactly this—moving the parentheses around without changing the numbers or operation.

Why the others are wrong:
- B shows the *commutative* law (order doesn't matter), not associative (grouping).
- C mixes addition and multiplication—not a valid property, and the sides aren't even equal.
- D also mixes operations and isn't equal on both sides.

Why A is correct:
The associative law allows you to regroup terms, but it doesn't let you rearrange them. Option A moves the 3 and 2 around (it's now $1 + (3 + 2)$ instead of $1 + (2 + 3)$), which violates the associative law. This actually uses the *commutative* law.

Why the others are wrong:
- B: Correctly regroups without rearranging: $(1 + 2) + 3 = 1 + (2 + 3)$ ✓
- C: Correctly regroups multiplication: $(3 × 2) × 4 = 3 × (2 × 4)$ ✓
- D: Correctly regroups addition: $(5 + 6) + 1 = 5 + (6 + 1)$ ✓

Why A is correct:
The associative law lets you regroup numbers in addition without changing the result. Starting with (x + y) + z, you can regroup it as x + (y + z), which gives 2 + (3 + 4) = 2 + 7 = 9. This directly shows the associative property being applied.

Why the others are wrong:
- B: While it equals 9, this rearranges the *order* of numbers (moving z before y), which is the commutative law, not associative.
- C: This just adds the numbers without showing the associative property being used—it's not a demonstration of the law.
- D: This is just the original expression solved (2 + 3 + 4 = 9), but it equals 9, not 10, so it's mathematically incorrect anyway.

• Left side: (2 × 3) × 4 = 6 × 4 = 24
• Right side: 2 × (3 × 4) = 2 × 12 = 24

Both sides equal 24, which demonstrates the associative property of multiplication—it doesn't matter how you group the numbers, you get the same answer.

Why others are wrong:
- B, C, D are all smaller numbers that don't equal either side of the equation.

Correct Answer (A): The associative law states that when you change how you *group* numbers with parentheses—like (2+3)+4 versus 2+(3+4)—you get the same result. This works for both addition and multiplication.

Why others are wrong:
- B: This describes the commutative law (order doesn't matter), not associative (grouping doesn't matter). Changing order actually *does* change results in some operations.
- C: This is the distributive law, which involves multiplying across addition (like 2(3+4) = 2·3 + 2·4).
- D: This is the identity property of addition, not associative.

Multiplication is associative because (a × b) × c = a × (b × c). No matter how you group the numbers, you get the same result. For example: (2 × 3) × 4 = 6 × 4 = 24, and 2 × (3 × 4) = 2 × 12 = 24.

Subtraction is NOT associative: (10 − 5) − 2 = 3, but 10 − (5 − 2) = 7. Different results!

Division is NOT associative: (12 ÷ 4) ÷ 2 = 1.5, but 12 ÷ (4 ÷ 2) = 6. Different results!

So D is wrong because we found an associative operation (multiplication).

A is correct because it only multiplies the first number by 2, leaving the 4 unchanged. The distributive law requires multiplying *both* terms inside the parentheses by 2.

B is correct — this is the proper statement of the distributive law: multiply each term by 2.

C is correct — this correctly applies the distributive law and simplifies: 2×3 + 2×4 = 6 + 8.

D is correct — this is the final simplified answer: 2(7) = 14.

Correct Answer: A ($7x - 14$)

The distributive law says multiply the number outside the parentheses by *each* term inside: $7 \times x = 7x$ and $7 \times (-2) = -14$. So $7(x - 2) = 7x - 14$.

Why others are wrong:
- B ($x - 14$): Forgot to multiply the $x$ by 7
- C ($7x - 2$): Multiplied 7 by $x$ correctly, but forgot to multiply 7 by the $-2$
- D ($7x + 14$): Got the multiplication right but made the sign wrong—should be minus, not plus

Correct Answer: A ($14$)

• Work inside the parentheses first: (2 + 3) = 5 and (4 + 5) = 9
• Then add them together: 5 + 9 = 14

Why the others are wrong:
- B ($15$): This might come from miscalculating 4 + 5 as 10, then 5 + 10 = 15
- C ($12$): This could result from incorrectly adding only some of the numbers
- D ($13$): This is close but comes from an arithmetic error in the final step

A is correct: The distributive law says multiply the number outside the parentheses by each term inside: 3 × 5 = 15 and 3 × x = 3x, giving 15 + 3x.

B is wrong: You can't just add 3 + 5 to get 8 and then multiply by x—that ignores the distributive property.

C is wrong: This reverses the coefficients and loses the constant term entirely.

D is wrong: This has no connection to distributing 3 across both terms in the parentheses.

A is correct: The commutative law says you can rearrange the *order* of numbers being added without changing the sum. Here, the terms shift from $a + b + c$ to $c + b + a$—same numbers, different order, same result.

B is wrong: This shows the *associative* law (changing which terms you group with parentheses), not commutative.

C is wrong: This is just the reflexive property (something equals itself)—no rearrangement happens, so it doesn't demonstrate commutativity.

D is wrong: This is the *distributive* law (multiplication distributed over addition), not commutative.

Why A is correct:
The distributive law means finding the greatest common factor (GCF) of all terms. Both 8x and 12 share a GCF of 4, so you factor it out: 4(2x + 3). Check: 4 × 2x + 4 × 3 = 8x + 12 ✓

Why the others are wrong:
- B: 2(4x + 12) only factors out 2, not the GCF. You could simplify it further to 4(2x + 3).
- C: While 8(x + 1.5) is technically correct, it's not proper factoring—we don't factor out 8 when 4 is the GCF, and we avoid decimals when possible.
- D: This isn't factored at all; it's just two terms added together.

Subtraction is not commutative because the order matters. For example, 5 − 3 = 2, but 3 − 5 = −2. You get different answers.

Why the others are wrong:
- Addition: Commutative (5 + 3 = 3 + 5 = 8)
- Multiplication: Commutative (5 × 3 = 3 × 5 = 15)
- None of the above: Wrong because subtraction *is* a non-commutative operation

A is correct because it shows both laws:
- The associative law appears in the middle step: $2(3 + (4 + 5))$ regroups the addition without changing the result
- The distributive law appears in the final step: multiplying 2 by each term inside the parentheses

B is wrong — this only shows the commutative law (reordering), not associative or distributive.

C is wrong — this shows the commutative law for multiplication, not associative or distributive.

D is wrong — this only shows the associative law for addition; there's no distributive property involved.

# Explanation

A is correct: The associative law says you can regroup numbers when multiplying without changing the result. Both sides equal 30, showing that (2 × 3) × 5 = 6 × 5 and 2 × (3 × 5) = 2 × 15, proving the grouping doesn't matter.

B is wrong: This shows the *commutative* law (order doesn't matter), not the associative law. It's also addition, not multiplication.

C is wrong: This demonstrates the *distributive* law (multiplication distributed over addition), not the associative law.

D is wrong: This is just a basic multiplication fact, not any property law.

A. $3x + 12$ ✓

The distributive law says you multiply the number outside the parentheses by *each* term inside: $3(x + 4) = 3 \cdot x + 3 \cdot 4 = 3x + 12$.

Why the others are wrong:
- B ($x + 12$): You forgot to multiply the $x$ by 3—you only multiplied the 4.
- C ($3x + 4$): You multiplied $x$ by 3 correctly, but forgot to multiply the 4 by 3.
- D ($3 + 4x$): This reverses the problem—you multiplied incorrectly and in the wrong order.

Correct Answer: A. $7 + (3 + 2)$

The associative law lets you regroup numbers being added without changing the result. Here, we moved the parentheses from $(7 + 3) + 2$ to $7 + (3 + 2)$—same numbers, same operation, just grouped differently.

Why the others are wrong:
- B. This changes the *order* of numbers (commutative law), not just their grouping
- C. This introduces multiplication, which breaks the original problem
- D. This solves the problem instead of just regrouping it using the associative law

Correct answer: A ($4x - 20$)

The distributive law says multiply the number outside the parentheses by *each* term inside: $4 \times x = 4x$ and $4 \times (-5) = -20$, giving $4x - 20$.

Why the others are wrong:
- B ($4x + 20$): Wrong sign—you'd get a plus if you accidentally multiplied $4 \times 5$ instead of $4 \times (-5)$.
- C ($x - 20$): You forgot to multiply the $x$ by 4; you only distributed to the second term.
- D ($4x - 5$): You multiplied only the 4 by $x$ but forgot to multiply the 4 by the $-5$.

# Explanation

Why A is correct:
This uses *both* laws together. First, the associative law groups $(4 + 1)$ inside the parentheses. Then the distributive law multiplies 2 by each term: $2(3 + 5) = 2×3 + 2×5$ (which equals $2×3 + 2×4 + 2×1$). It demonstrates both properties working.

Why the others are wrong:
- B: Only shows the commutative law (reordering), not associative or distributive.
- C: Only shows the commutative law for multiplication, not either required law.
- D: Only shows the associative law for addition. It doesn't use the distributive law at all.

Why A is correct:
The Distributive Law states that a × (b + c) = a × b + a × c. Here, 5 is "distributed" across the addition (2 + 3), which is exactly what the equation shows. The answer is 25 because 5 × 5 = 25 (or 10 + 15 = 25).

Why the others are wrong:
- B (Associative): The Associative Law deals with grouping, like (a + b) + c = a + (b + c)—there's no regrouping here.
- C (Commutative): The Commutative Law is about order, like a × b = b × a—the order of numbers doesn't change here.
- D: While this correctly names the Distributive Law, the answer 20 is wrong (5 × 5 = 25, not 20).

Correct answer: A. $(6 + 4) + 5$

The associative law lets you regroup numbers being added without changing the result. Starting with $6 + (4 + 5)$, you move the parentheses to group the first two numbers instead: $(6 + 4) + 5$. Both equal 15, but the grouping changes.

Why the others are wrong:
- B. Changes the order of numbers (uses commutative law, not associative)
- C. Still groups 4 and 5 together—this doesn't actually regroupify anything
- D. This is the final numerical answer, not a regrouping using the associative law

Correct Answer: A

The commutative law says you can swap the order of numbers when adding or multiplying, and get the same result. In $8 + 2 = 2 + 8$, the numbers switch places but the sum stays 10—this is commutative.

Why the others are wrong:

• B shows the associative law (regrouping with parentheses, not reordering)
• C shows the distributive law (multiplying across a sum)
• D shows the identity property (adding zero doesn't change the number)

A. $9(2x + 3)$ is correct because 9 is the greatest common factor (GCF) of 18 and 27, and when you distribute it back: $9 × 2x + 9 × 3 = 18x + 27$ ✓

B. $3(6x + 9)$ uses a common factor, but 3 is not the *greatest* common factor—you can factor out more.

C. $6(3x + 4.5)$ has a common factor of 6, but it's smaller than the GCF, and introduces an unnecessary decimal.

D. $18(x + 1.5)$ uses 18 as a factor, but 27 ÷ 18 = 1.5, creating a decimal—factoring should give whole numbers inside the parentheses when possible.

Why A is correct:
The distributive law means multiply 4 by each term inside the parentheses: 4 × x = 4x and 4 × 3 = 12, giving you 4x + 12. Then combine with 2x: 4x + 2x + 12 = 6x + 12.

Why the others are wrong:
- B: This shows the distribution step but forgets to combine like terms (4x + 2x should equal 6x, not stay separate).
- C: This skips the distribution entirely—you can't just remove the parentheses without multiplying by 4.
- D: This has the right terms combined (6x) but lost the constant (should be +12, not +3).

A. $5a$ ✓

When adding like terms (terms with the same variable), add the coefficients: 2 + 3 = 5, so 2a + 3a = 5a.

Why the others are wrong:
- B. $6a$ – Multiplying instead of adding: 2 × 3 = 6 (incorrect operation)
- C. $5a^2$ – You only square the variable if you're multiplying terms like a × a, not adding them
- D. $a + a$ – This doesn't simplify the original expression and equals 2a, not the answer

A. Addition ✓

Addition is commutative (order doesn't matter: 3 + 5 = 5 + 3) and associative (grouping doesn't matter: (2 + 3) + 4 = 2 + (3 + 4)).

Why the others are wrong:
- B. Subtraction: Not commutative (5 − 3 ≠ 3 − 5) and not associative ((10 − 5) − 2 ≠ 10 − (5 − 2))
- C. Division: Not commutative (8 ÷ 2 ≠ 2 ÷ 8) and not associative ((12 ÷ 6) ÷ 2 ≠ 12 ÷ (6 ÷ 2))
- D. None of the above: Incorrect because addition does have both properties

Why A is correct:
Both terms have (x + y) as a common factor, so you can factor it out: 2(x + y) + 3(x + y) = (2 + 3)(x + y) = 5(x + y).

Why the others are wrong:
- B: 6x + 6y would mean 2x + 2y + 3x + 3y, but that's not what we have here—we're adding 2(x + y) + 3(x + y), not separate terms.
- C: While 5x + 5y is equivalent to 5(x + y), it's a less simplified form—the factored form A is the better answer.
- D: This doesn't follow the order of operations correctly and would only give 5x + y, which is wrong.

A is correct. The distributive law means you multiply the number outside the parentheses (5) by each term inside: 5 × x = 5x, and 5 × 2y = 10y, giving you 5x + 10y.

B is wrong because you forgot to multiply the 2y by 5; you only distributed to the x.

C is wrong because you multiplied 5 by 2y correctly, but didn't multiply 5 by x.

D is wrong because there's no reason to add a "+5" at the end; you only distribute to terms inside the parentheses.

# Explanation

A is correct. The commutative law of multiplication states that you can swap the order of factors and get the same product: $a × b = b × a$. This shows exactly that: $7 × 4 = 4 × 7$ (both equal 28).

B is wrong. This demonstrates the distributive property or breaking apart a number, not the commutative law. It shows $7 × 4$ equals two groups of $7 × 2$, but doesn't involve swapping factors.

C is wrong. This shows the associative law of multiplication, which is about regrouping factors with parentheses: $(a × b) × c = a × (b × c)$. It's not about swapping the order.

D is wrong. This is incomplete/incorrect application of the distributive property. The right form would be $7 × (4 + 2) = (7 × 4) + (7 × 2)$, and the equation given doesn't equal the same value on both sides anyway.

A is correct because it shows subtraction is NOT associative—the two sides actually give different results: (10 - 2) - 3 = 5, but 10 - (2 - 3) = 11.

B is wrong because addition IS associative—both sides equal 9, so this is valid.

C is wrong because multiplication IS associative—both sides equal 30, so this is valid.

D is wrong because addition IS associative—both sides equal 10, so this is valid.

The question asks which is "not valid due to non-associativity," meaning which operation fails the associative property. Only subtraction fails it here.

A is correct: The associative law lets you regroup terms without changing their order—it only changes where the parentheses go. Moving the parentheses from (a + b) + (c + d) to a + (b + c) + d follows this rule perfectly.

B is wrong: This rearranges the *order* of terms (a and c swap positions), which violates the associative law—that's the commutative law.

C is wrong: This adds an extra "+1" that wasn't in the original expression, so it changes the value.

D is wrong: This duplicates terms (a and b appear twice), creating a completely different expression.

A is correct because the distributive law says you can multiply a number by a sum by distributing it to each term: 3(x + y) = 3x + 3y.

B is wrong—that's the commutative law (order doesn't matter in addition).

C is wrong—that's the associative law (grouping doesn't matter in addition).

D is wrong—that's the identity property of addition (adding zero doesn't change a number).

Correct answer: A ($2x+16$)

First, simplify inside the parentheses: 5 + x + 3 = x + 8. Then distribute the 2: 2(x + 8) = 2x + 16. ✓

• B ($2x + 8$): This forgets to distribute the 2 to all terms—you'd only get 2x + 8 if you multiplied 2 by x and 4, not by 8.
• C ($2x + 15$): This incorrectly multiplies 2 × (5 + 3) = 10, then adds the original 5 instead of the result.
• D ($2x + 10$): This only multiplies 2 by the numbers (5 + 3 = 8, but 2 × 5 = 10), missing 2 × x in the distribution.

A is correct: The identity property of multiplication states that any number multiplied by 1 equals itself. The number 1 is called the "multiplicative identity" because it leaves numbers unchanged.

Why the others are wrong:
- B shows the identity property of *addition* (0 is the additive identity), not multiplication
- C shows the *commutative* property of addition (order doesn't matter)
- D shows the *associative* property of addition (grouping doesn't matter)

Why A is correct:
Combine like terms step by step: first (2x + 3x) = 5x, then add the remaining 4x to get 5x + 4x = 9x.

Why the others are wrong:
- B (6x): Only adds 2x + 3x and ignores the 4x at the end
- C (5x): Only solves the part in parentheses and forgets to add 4x
- D (8x): Likely miscounted or added incorrectly instead of combining all three terms

Correct answer: A. $4x + 15$

• Distribute the 3: $3(2x + 5) = 6x + 15$
• Subtract 2x: $6x + 15 - 2x = 4x + 15$ ✓

Why others are wrong:
- B ($6x + 5$): You forgot to distribute the 3 to the 5, or made an error combining like terms
- C ($5x + 15$): This combines $6x - 2x$ incorrectly (should be $4x$, not $5x$)
- D ($4x + 10$): You correctly got $4x$ but miscalculated the constant (should be $+15$, not $+10$)

Why A is correct:
Distribute the 2 into the parentheses: 2(x + 3) = 2x + 6. Then combine like terms: 5x + 2x + 6 = 7x + 6.

Why the others are wrong:
- B: This shows the steps before combining like terms—it's not simplified yet.
- C: The 6 was incorrectly treated as a variable term (6x) instead of a constant.
- D: The constant should be 6, not 5 (from 2 × 3).

A is correct because the commutative property states that the *order* of numbers being added doesn't change the sum—you can swap them around and get the same result.

B is wrong—this is the *associative* property, which groups numbers differently (with parentheses) rather than reordering them.

C is wrong—this is the *distributive* property, which involves multiplication and addition together.

D is wrong—this is the *identity* property of addition, which shows that adding zero doesn't change a number.

A is correct because 3 is the greatest common factor (GCF) of all three terms (6x, 9y, and 3z). When you factor out 3, you get 3(2x + 3y - z), and multiplying back confirms it works: 3·2x + 3·3y + 3·(-z) = 6x + 9y - 3z. ✓

B is wrong because while it does equal the original expression, 6 is not the GCF—we can factor out 3, not 6.

C is wrong because the sign on z is incorrect (it should be negative, not positive).

D is wrong because it's not factored at all—it's just a different expression that doesn't equal 6x + 9y - 3z.

Why A is correct:
The associative law only works for addition and multiplication, not subtraction. When you change where parentheses go in subtraction, you get different answers: $(5 - 2) - 1 = 2$ but $5 - (2 - 1) = 4$. This correctly shows that associative law does NOT apply here.

Why the others are wrong:
- B: This correctly applies associative law to addition—both sides equal 9.
- C: This correctly applies associative law to multiplication—both sides equal 12.
- D: This is the general associative property formula for addition, which is always true.

A is correct. Use the distributive property: multiply 5 by each term inside the parentheses.
- 5 × a = 5a
- 5 × 2b = 10b
- 5 × (−3c) = −15c

B is wrong: You forgot to multiply the terms by 5; you just rewrote the expression.

C is wrong: You multiplied incorrectly—the last term should be negative (−15c), not positive.

D is wrong: This doesn't follow the distributive property; you can't just add coefficients together.