This question tests using properties of operations to generate equivalent expressions: distributive (expand/factor), commutative (reorder), associative (regroup), combining like terms. Properties: distributive a(b+c)=ab+ac (expand: 3(2+x)=6+3x multiply 3 to each term, or factor: 24x+18y=6(4x+3y) pull out GCF=6), commutative a+b=b+a (order doesn't matter: x+5=5+x), combining like terms ax+bx=(a+b)x (same variable combines: y+y+y=1y+1y+1y=3y). Application: expand by distributing (3 to 2 and x), factor by finding GCF (24 and 18 have GCF 6, divide each: 24x/6=4x, 18y/6=3y, write 6(4x+3y)). Example: expand 3(2+x) by distributing: 3×2=6, 3×x=3x, result 6+3x; or factor 24x+18y: find GCF (factors of 24: 1,2,3,4,6,8,12,24; factors of 18: 1,2,3,6,9,18; common: 6 is greatest), factor out: 6(24x/6+18y/6)=6(4x+3y); or combine y+y+y=(1+1+1)y=3y. Here, the correct equivalent expression using the distributive property to expand 3(2x + 5) is 6x + 15, by multiplying 3 by 2x and 3 by 5. A common error is incomplete distribution, like 3(2x + 5)=6x + 5 missing the 15, or arithmetic wrong like 3×5=8. Expanding: distribute multiplier to every term inside parentheses (a(b+c)=ab+ac, don't miss any terms). Mistakes: distribution errors most common at grade 6, GCF identification wrong, combining unlike terms, arithmetic errors.

This question tests using properties of operations to generate equivalent expressions, combining commutative property and like terms. First, use commutative property to reorder 6x + 2y + 4x as 6x + 4x + 2y (order doesn't matter), then combine like terms: (6 + 4)x + 2y = 10x + 2y. For example, reordering groups the x terms: 6x + 4x = 10x, with 2y unchanged, resulting in 10x + 2y. The correct equivalent expression using these properties is 10x + 2y. A common error is combining unlike terms, like adding x and y to 12xy, or misadding coefficients. Apply commutative to reorder, then combine only same variables by adding coefficients. To verify equivalence, test with x = 1, y = 1: 6(1) + 2(1) + 4(1) = 12 and 10(1) + 2(1) = 12, which matches.

This question tests using properties of operations to generate equivalent expressions, specifically combining like terms. Combining like terms involves adding coefficients of similar terms: 2x + 3x + 5 = (2 + 3)x + 5 = 5x + 5, where x terms combine but the constant 5 stays separate. For example, in 2x + 3x + 5, group the like terms 2x + 3x = 5x, then add the constant: 5x + 5. The correct equivalent expression by combining like terms is 5x + 5. A common error is combining unlike terms, like adding 5 to 5x to make 6x + 5, or squaring to $5x^2$ + 5. When combining like terms, only add coefficients for terms with the same variable, leaving constants alone. To verify equivalence, test with x = 1: 2(1) + 3(1) + 5 = 10 and 5(1) + 5 = 10, which matches; errors like 5x give 5, which doesn't.

This question tests using properties of operations to generate equivalent expressions, specifically the distributive property to expand 3(2x+y). The distributive property: a(b+c)=ab+ac, so multiply 3 by each: 3×2x=6x, 3×y=3y, result 6x+3y. For example, expand 3(2+x): 3×2=6, 3×x=3x, 6+3x. The correct equivalent expression using distributive property is 6x+3y. A common error is incomplete distribution, like 3(2x+y)=6x+y missing 3y, or wrong like 3x+2y. Expanding: distribute to every term, a(b+c)=ab+ac, don't miss terms. Distribution errors are most common at grade 6.

This question tests using properties of operations to generate equivalent expressions: distributive (expand/factor), commutative (reorder), associative (regroup), combining like terms. Properties: distributive a(b+c)=ab+ac (expand: 3(2+x)=6+3x multiply 3 to each term, or factor: 24x+18y=6(4x+3y) pull out GCF=6), commutative a+b=b+a (order doesn't matter: x+5=5+x), combining like terms ax+bx=(a+b)x (same variable combines: y+y+y=1y+1y+1y=3y). Application: expand by distributing (3 to 2 and x), factor by finding GCF (24 and 18 have GCF 6, divide each: 24x/6=4x, 18y/6=3y, write 6(4x+3y)). Example: expand 3(2+x) by distributing: 3×2=6, 3×x=3x, result 6+3x; or factor 24x+18y: find GCF (factors of 24: 1,2,3,4,6,8,12,24; factors of 18: 1,2,3,6,9,18; common: 6 is greatest), factor out: 6(24x/6+18y/6)=6(4x+3y); or combine y+y+y=(1+1+1)y=3y. Here, the correct equivalent expression by factoring 18x + 12y completely using the GCF of 6 is 6(3x + 2y). A common error is using a wrong GCF like 3, resulting in 3(6x + 4y), which is not completely factored since 6 and 4 share a further factor of 2. Factoring: (1) find GCF of all coefficients (24 and 18: list factors, identify greatest common: 6), (2) divide each term by GCF (24x÷6=4x, 18y÷6=3y), (3) write GCF(quotients): 6(4x+3y). Mistakes: distribution errors most common at grade 6, GCF identification wrong, combining unlike terms, arithmetic errors.

This question tests using properties of operations to generate equivalent expressions: distributive (expand/factor), commutative (reorder), associative (regroup), combining like terms. Properties: distributive a(b+c)=ab+ac (expand: 3(2+x)=6+3x multiply 3 to each term, or factor: 24x+18y=6(4x+3y) pull out GCF=6), commutative a+b=b+a (order doesn't matter: x+5=5+x), combining like terms ax+bx=(a+b)x (same variable combines: y+y+y=1y+1y+1y=3y). Application: expand by distributing (3 to 2 and x), factor by finding GCF (24 and 18 have GCF 6, divide each: 24x/6=4x, 18y/6=3y, write 6(4x+3y)). Example: expand 3(2+x) by distributing: 3×2=6, 3×x=3x, result 6+3x; or factor 24x+18y: find GCF (factors of 24: 1,2,3,4,6,8,12,24; factors of 18: 1,2,3,6,9,18; common: 6 is greatest), factor out: 6(24x/6+18y/6)=6(4x+3y); or combine y+y+y=(1+1+1)y=3y. Here, the correct equivalent expression by factoring 30x + 25 completely using the GCF of 5 is 5(6x + 5). A common error is using a wrong GCF like 25, resulting in 25(x + 1) but 30x/25 is not integer, or arithmetic wrong like dividing 30/10=3 but missing complete factoring. Factoring: (1) find GCF of all coefficients (24 and 18: list factors, identify greatest common: 6), (2) divide each term by GCF (24x÷6=4x, 18y÷6=3y), (3) write GCF(quotients): 6(4x+3y). Mistakes: distribution errors most common at grade 6, GCF identification wrong, combining unlike terms, arithmetic errors.

This question tests using properties of operations to generate equivalent expressions, specifically the distributive property to expand expressions. The distributive property states that a(b + c) = ab + ac, so to expand 3(x + 2), multiply 3 by each term inside the parentheses: 3 × x = 3x and 3 × 2 = 6, resulting in 3x + 6. For example, expanding 3(x + 2) gives 3x + 6, which represents the cost of x notebooks at $3 each plus a $2 folder, as 3x for notebooks and 6 more (which is 3 × 2) for the folders if misinterpreted, but correctly it's 3x + 6. The correct equivalent expression using the distributive property is 3x + 6. A common error is incomplete distribution, like writing 3x + 2 instead of multiplying 3 by 2 to get 6, or doubling incorrectly to 6x + 2. When expanding, always distribute the multiplier to every term inside the parentheses, so a(b + c) = ab + ac, without missing any terms. To verify equivalence, test with a value like x = 1: 3(1 + 2) = 9 and 3(1) + 6 = 9, which matches.

This question tests using properties of operations to generate equivalent expressions, specifically combining like terms in 7y-2y+4. Combining like terms: 7y-2y=(7-2)y=5y, constant +4 remains, so 5y+4. For example, combine 2x+3x+5=5x+5. The correct equivalent expression by combining like terms is 5y+4. A common error is wrong subtraction, like 7-2=9 leading to 9y+4, or combining unlike terms like 5y+4=9y. Combining like terms: only same variables, (7-2)y+4=5y+4, not y with constants. Arithmetic errors in coefficients are common at grade 6.

This question tests using properties of operations to generate equivalent expressions: distributive (expand/factor), commutative (reorder), associative (regroup), combining like terms. Properties: distributive a(b+c)=ab+ac (expand: 3(2+x)=6+3x multiply 3 to each term, or factor: 24x+18y=6(4x+3y) pull out GCF=6), commutative a+b=b+a (order doesn't matter: x+5=5+x), combining like terms ax+bx=(a+b)x (same variable combines: y+y+y=1y+1y+1y=3y). Application: expand by distributing (3 to 2 and x), factor by finding GCF (24 and 18 have GCF 6, divide each: 24x/6=4x, 18y/6=3y, write 6(4x+3y)). Example: expand 3(2+x) by distributing: 3×2=6, 3×x=3x, result 6+3x; or factor 24x+18y: find GCF (factors of 24: 1,2,3,4,6,8,12,24; factors of 18: 1,2,3,6,9,18; common: 6 is greatest), factor out: 6(24x/6+18y/6)=6(4x+3y); or combine y+y+y=(1+1+1)y=3y. Here, the correct equivalent expression by combining like terms in 4y + 3 + 2y is 6y + 3, adding the y terms (4 + 2 = 6) and leaving the constant 3. A common error is combining unlike terms, like adding 6y + 3 to get 9y, or leaving it as 4y + 5y. Combining like terms: 2x+3x+5=(2+3)x+5=5x+5 (only combine same variables, not x with constants). Mistakes: distribution errors most common at grade 6, GCF identification wrong, combining unlike terms, arithmetic errors.

This question tests using properties of operations to generate equivalent expressions, specifically factoring by finding the greatest common factor (GCF). The distributive property in reverse allows factoring: for 24x + 18y, find the GCF of 24 and 18, which is 6, then divide each term by 6 (24x ÷ 6 = 4x, 18y ÷ 6 = 3y) and write 6(4x + 3y). For example, to factor 24x + 18y, list factors of 24 (1, 2, 3, 4, 6, 8, 12, 24) and 18 (1, 2, 3, 6, 9, 18), identify the greatest common as 6, then factor out: 6(4x + 3y). The correct equivalent expression using factoring with GCF is 6(4x + 3y). A common error is choosing a smaller common factor like 2 or 3 instead of the greatest 6, or misdividing terms like writing 6(3x + 4y) which equals 18x + 24y. When factoring, first find the GCF of all coefficients by listing factors and picking the largest common one, then divide each term accurately. To verify equivalence, test with x = 1, y = 1: 24(1) + 18(1) = 42 and 6(4(1) + 3(1)) = 6(7) = 42, which matches.