Which expression is equivalent to when simplified completely?
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7th Grade Math Quiz
Practice Add Subtract Factor Linear Expressions in 7th Grade Math with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.
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Which expression is equivalent to 21(6x−8)+43(4x+12) when simplified completely?
This quiz focuses on Add Subtract Factor Linear Expressions, giving you a quick way to practice the rules, question types, and explanations that matter most for 7th Grade Math.
Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.
Which expression is equivalent to 21(6x−8)+43(4x+12) when simplified completely?
Explanation: Distribute each fraction: 21(6x−8)+43(4x+12)=3x−4+3x+9=6x+5. Choice B results from calculating −4+9=4 instead of 5. Choice C comes from incorrectly adding 3x+3x=5x. Choice D results from errors in both distribution and combining terms.
When subtracting 2(3x−5) from 4(2x+3), what is the coefficient of x in the simplified result?
Explanation: Subtracting 2(3x−5) from 4(2x+3) means: 4(2x+3)−2(3x−5)=8x+12−6x+10=2x+22. The coefficient of x is 2. Choice B comes from adding instead of subtracting: 8x+6x=14x. Choice C results from calculating 6x−8x=−2x. Choice D comes from incorrectly distributing.
A student factored the expression 12x+18y−6 and wrote 6(2x+3y−1). To verify this is correct, which property should be applied?
Explanation: To verify factoring is correct, use the distributive property to expand 6(2x+3y−1)=12x+18y−6, confirming it matches the original expression. Choice A (commutative) deals with order, choice B (associative) deals with grouping, and choice D (identity) is about value preservation but doesn't describe the verification method.
The expression ax+bx+c can be factored as x(a+b)+c when certain conditions are met. Which statement about this factoring is most accurate?
Explanation: When you encounter factoring problems, remember that factoring means rewriting an expression as a product of simpler terms. The key is understanding when factoring is correct versus when it's complete. The expression ax+bx+c can indeed be factored as x(a+b)+c because you're pulling out the common factor x from the first two terms. This is mathematically valid regardless of the values of a, b, or c. However, this doesn't mean you've found the greatest common factor of all terms. Let's examine why the other options miss the mark. Choice A incorrectly claims this only works when c=0. While having c=0 would allow complete factoring as x(a+b), the partial factoring x(a+b)+c is still mathematically correct when c=0. Choice B gets the relationship backwards—when c=0, you can factor completely, making the factoring more complete, not less. Choice C focuses on the coefficient a+b=1, but this restriction isn't necessary for the factoring to be valid. Choice D correctly identifies that this factoring is always mathematically sound but acknowledges it may not represent the most simplified form. For instance, if all terms share a common factor, you haven't fully factored until you extract that greatest common factor. Study tip: When factoring, always check if you can pull out more common factors after your first step. Complete factoring means finding the greatest common factor of all terms, not just some of them.
When factoring 21a−14b+35c, a student writes 7(3a−2b+5c). What should the student do to verify this factorization is correct?
Explanation: To verify any factorization, expand it using the distributive property: 7(3a−2b+5c)=21a−14b+35c, which matches the original expression. While choices A, B, and D describe properties of correct factorization, choice C describes the direct verification method that confirms the factorization is mathematically equivalent to the original expression.
Sara needs to add 3(x−2)+5x and 2(3−x)−4. If she first expands each expression completely before combining, what will be her final simplified answer?
Explanation: When you encounter expressions that need to be added together, the key is to expand each expression first using the distributive property, then combine like terms systematically. Let's expand the first expression: 3(x−2)+5x. Using the distributive property, 3(x−2)=3x−6, so this becomes 3x−6+5x=8x−6. For the second expression: 2(3−x)−4. Distributing gives us 2(3−x)=6−2x, so this becomes 6−2x−4=2−2x. Now we add the expanded expressions: (8x−6)+(2−2x). Combining like terms: 8x−2x−6+2=6x−4. Looking at the wrong answers: Choice A (6x−6) likely comes from incorrectly combining the constant terms as −6−4=−10 instead of −6+2=−4. Choice B (6x+2) results from sign errors when distributing or combining terms. Choice C (8x−4) happens when you correctly handle the constants but forget to combine the x terms: 8x−2x=6x, not 8x. The correct answer is D: 6x−4. Strategy tip: Always expand completely before combining, and double-check your signs when distributing negative terms. Work step-by-step rather than trying to do multiple operations mentally—this prevents the sign errors that create most wrong answers in these problems.
Combine like terms to simplify (2x−43)+(−5x+21).
Explanation: This question tests adding, subtracting, factoring, and expanding linear expressions with rational coefficients using properties of operations. Operations: add by combining like terms (3x+2x=5x, coefficients add), subtract by distributing negative then combining (4x-(2x-3)=4x-2x+3=2x+3, negative distributes to all terms), expand using distributive property a(b+c)=ab+ac (multiply each term: 3(2x-5)=6x-15), factor by finding GCF and dividing out (6x+9: GCF=3, so 3(6x/3+9/3)=3(2x+3)). Combining like terms requires same variable (3x and 5x combine, but 2x and 3 don't). For example, (2x - 3/4) + (-5x + 1/2) combines x terms (2 - 5 = -3x) and constants (-3/4 + 2/4 = -1/4), giving -3x - 1/4. In this case, the correct simplification is -3x - 1/4. A common error is sign error in adding fractions without a common denominator, such as -3/4 + 1/2 = -3/4 + 1/2 = 1/4 instead of -1/4. Strategy: (1) for adding/subtracting: distribute any negatives first (important for subtraction), identify like terms (same variable part: 3x and 5x are like, 2x and 3 aren't), combine (add/subtract coefficients: 3x+5x=8x), combine constants separately (5-3=2); (2) for expanding: distribute to every term (a(b+c+d)=ab+ac+ad, don't miss any); (3) for factoring: find GCF of all terms, divide each term by GCF, write as GCF(quotients). Common mistakes: distributing negative to only first term -(2x-3)=-2x-3 (wrong, should be -2x+3), combining unlike terms, fraction operations without common denominators, factoring incompletely.
A student simplifies an expression for points earned in a game: 12x+18. Factor the expression completely by finding the greatest common factor (GCF).
Explanation: This question tests adding, subtracting, factoring, and expanding linear expressions with rational coefficients using properties of operations. Operations include adding by combining like terms (3x+2x=5x, coefficients add), subtracting by distributing the negative then combining (4x-(2x-3)=4x-2x+3=2x+3, negative distributes to all terms), expanding using the distributive property a(b+c)=ab+ac (multiply each term: 3(2x-5)=6x-15), and factoring by finding the GCF and dividing out (6x+9: GCF=3, so 3(6x/3+9/3)=3(2x+3)). Combining like terms requires the same variable (3x and 5x combine, but 2x and 3 don't). For example, (3x+5)+(2x-3) combines like terms: 3x+2x=5x, 5+(-3)=2, giving 5x+2; or (4x+7)-(2x+3) distributes the negative: 4x+7-2x-3, combines to 2x+4; or factor 8x+12: GCF=4, so 4(2x+3). Factor completely by finding GCF 6: 12x/6 = 2x, 18/6 = 3, giving 6(2x + 3), which is choice A. A common error is factoring partially, like pulling out only 3 to get 3(4x + 6) without checking for a larger GCF. Strategy: (1) for adding/subtracting: distribute any negatives first (important for subtraction), identify like terms (same variable part: 3x and 5x are like, 2x and 3 aren't), combine (add/subtract coefficients: 3x+5x=8x), combine constants separately (5-3=2); (2) for expanding: distribute to every term (a(b+c+d)=ab+ac+ad, don't miss any); (3) for factoring: find GCF of all terms, divide each term by GCF, write as GCF(quotients). Common mistakes: distributing negative to only first term -(2x-3)=-2x-3 (wrong, should be -2x+3), combining unlike terms, fraction operations without common denominators, factoring incompletely.
Simplify the difference: (1.2x−4.8)−(0.7x+1.5).
Explanation: This question tests adding, subtracting, factoring, and expanding linear expressions with rational coefficients using properties of operations. Operations include adding by combining like terms (3x+2x=5x, coefficients add), subtracting by distributing the negative then combining (4x-(2x-3)=4x-2x+3=2x+3, negative distributes to all terms), expanding using the distributive property a(b+c)=ab+ac (multiply each term: 3(2x-5)=6x-15), and factoring by finding the GCF and dividing out (6x+9: GCF=3, so 3(6x/3+9/3)=3(2x+3)). Combining like terms requires the same variable (3x and 5x combine, but 2x and 3 don't). A specific example is (4x+7)-(2x+3): distribute negative 4x+7-2x-3, combine 2x+4. For this problem, distribute negative: 1.2x - 4.8 - 0.7x - 1.5, combine x terms: 1.2x - 0.7x = 0.5x, constants: -4.8 - 1.5 = -6.3, giving 0.5x - 6.3. A common error is sign error in subtraction, like not distributing the negative to the constant term. Strategy: for adding/subtracting, distribute any negatives first (important for subtraction), identify like terms (same variable part: 3x and 5x are like, 2x and 3 aren't), combine (add/subtract coefficients: 3x+5x=8x), combine constants separately (5-3=2); common mistakes include distributing negative to only first term -(2x-3)=-2x-3 (wrong, should be -2x+3).
A music club sells two types of tickets. The money collected is modeled by (2/3 x + 5) dollars and (1/6 x - 2) dollars. Simplify the sum (2/3 x + 5) + (1/6 x - 2).
Explanation: Combine the x-terms: 2/3 x + 1/6 x = 4/6 x + 1/6 x = 5/6 x. Combine the constants: 5 + (-2) = 3. Adding these gives 5/6 x + 3, matching choice A. A common mistake is adding 2/3 and 1/6 without a common denominator, which can lead to an answer like 3/9 x (choice C) instead of the correct 5/6 x.
A coach writes an expression for total practice time: 3(32x−5). Expand the expression using the distributive property.
Explanation: This question tests adding, subtracting, factoring, and expanding linear expressions with rational coefficients using properties of operations. Operations include adding by combining like terms (3x+2x=5x, coefficients add), subtracting by distributing the negative then combining (4x-(2x-3)=4x-2x+3=2x+3, negative distributes to all terms), expanding using the distributive property a(b+c)=ab+ac (multiply each term: 3(2x-5)=6x-15), and factoring by finding the GCF and dividing out (6x+9: GCF=3, so 3(6x/3+9/3)=3(2x+3)). Combining like terms requires the same variable (3x and 5x combine, but 2x and 3 don't). For example, (3x+5)+(2x-3) combines like terms: 3x+2x=5x, 5+(-3)=2, giving 5x+2; or (4x+7)-(2x+3) distributes the negative: 4x+7-2x-3, combines to 2x+4; or factor 8x+12: GCF=4, so 4(2x+3). Expand by distributing 3: 3*(2/3)x = 2x, and 3*(-5) = -15, resulting in 2x - 15, which is choice A. A common error is incomplete distribution, like multiplying only the first term and forgetting the second, giving just 2x without -15. Strategy: (1) for adding/subtracting: distribute any negatives first (important for subtraction), identify like terms (same variable part: 3x and 5x are like, 2x and 3 aren't), combine (add/subtract coefficients: 3x+5x=8x), combine constants separately (5-3=2); (2) for expanding: distribute to every term (a(b+c+d)=ab+ac+ad, don't miss any); (3) for factoring: find GCF of all terms, divide each term by GCF, write as GCF(quotients). Common mistakes: distributing negative to only first term -(2x-3)=-2x-3 (wrong, should be -2x+3), combining unlike terms, fraction operations without common denominators, factoring incompletely.
A coach says the total practice time can be written as 3(2x−5). Expand the expression using the distributive property.
Explanation: This question tests adding, subtracting, factoring, and expanding linear expressions with rational coefficients using properties of operations. Operations: add by combining like terms (3x+2x=5x, coefficients add), subtract by distributing negative then combining (4x-(2x-3)=4x-2x+3=2x+3, negative distributes to all terms), expand using distributive property a(b+c)=ab+ac (multiply each term: 3(2x-5)=6x-15), factor by finding GCF and dividing out (6x+9: GCF=3, so 3(6x/3+9/3)=3(2x+3)). Combining like terms requires same variable (3x and 5x combine, but 2x and 3 don't). For example, expanding 3(2x-5) using the distributive property gives 32x - 35 = 6x - 15. In this case, the correct expansion is 3(2x-5)=6x-15. A common error is missing the distribution to the second term or mishandling the sign, such as 3*2x -5 =6x-5 instead of multiplying the 5 by 3. Strategy: (1) for adding/subtracting: distribute any negatives first (important for subtraction), identify like terms (same variable part: 3x and 5x are like, 2x and 3 aren't), combine (add/subtract coefficients: 3x+5x=8x), combine constants separately (5-3=2); (2) for expanding: distribute to every term (a(b+c+d)=ab+ac+ad, don't miss any); (3) for factoring: find GCF of all terms, divide each term by GCF, write as GCF(quotients). Common mistakes: distributing negative to only first term -(2x-3)=-2x-3 (wrong, should be -2x+3), combining unlike terms, fraction operations without common denominators, factoring incompletely.
Simplify the expression 0.5x+2.5−0.3x+1.5 by combining like terms.
Explanation: This question tests adding, subtracting, factoring, and expanding linear expressions with rational coefficients using properties of operations. Operations: add by combining like terms (3x+2x=5x, coefficients add), subtract by distributing negative then combining (4x-(2x-3)=4x-2x+3=2x+3, negative distributes to all terms), expand using distributive property a(b+c)=ab+ac (multiply each term: 3(2x-5)=6x-15), factor by finding GCF and dividing out (6x+9: GCF=3, so 3(6x/3+9/3)=3(2x+3)). Combining like terms requires same variable (3x and 5x combine, but 2x and 3 don't). For example, 0.5x + 2.5 - 0.3x + 1.5 combines x terms (0.5 - 0.3 = 0.2x) and constants (2.5 + 1.5 = 4), giving 0.2x + 4. In this case, the correct simplification is 0.2x + 4. A common error is combining unlike terms or mishandling decimals, such as adding all numbers without grouping. Strategy: (1) for adding/subtracting: distribute any negatives first (important for subtraction), identify like terms (same variable part: 3x and 5x are like, 2x and 3 aren't), combine (add/subtract coefficients: 3x+5x=8x), combine constants separately (5-3=2); (2) for expanding: distribute to every term (a(b+c+d)=ab+ac+ad, don't miss any); (3) for factoring: find GCF of all terms, divide each term by GCF, write as GCF(quotients). Common mistakes: distributing negative to only first term -(2x-3)=-2x-3 (wrong, should be -2x+3), combining unlike terms, fraction operations without common denominators, factoring incompletely.
Which expression is equivalent to 5(x−52)+3x after simplifying?
Explanation: This question tests adding, subtracting, factoring, and expanding linear expressions with rational coefficients using properties of operations. Operations: add by combining like terms (3x+2x=5x, coefficients add), subtract by distributing negative then combining (4x-(2x-3)=4x-2x+3=2x+3, negative distributes to all terms), expand using distributive property a(b+c)=ab+ac (multiply each term: 3(2x-5)=6x-15), factor by finding GCF and dividing out (6x+9: GCF=3, so 3(6x/3+9/3)=3(2x+3)). Combining like terms requires same variable (3x and 5x combine, but 2x and 3 don't). For example, 5(x - 2/5) + 3x expands to 5x - 2 + 3x, then combines to 8x - 2. In this case, the correct simplification is 8x - 2. A common error is not distributing the 5 to the -2/5 term fully, such as leaving it as 5x - 2/5 + 3x. Strategy: (1) for adding/subtracting: distribute any negatives first (important for subtraction), identify like terms (same variable part: 3x and 5x are like, 2x and 3 aren't), combine (add/subtract coefficients: 3x+5x=8x), combine constants separately (5-3=2); (2) for expanding: distribute to every term (a(b+c+d)=ab+ac+ad, don't miss any); (3) for factoring: find GCF of all terms, divide each term by GCF, write as GCF(quotients). Common mistakes: distributing negative to only first term -(2x-3)=-2x-3 (wrong, should be -2x+3), combining unlike terms, fraction operations without common denominators, factoring incompletely.
Factor the expression 53x+56 by taking out the greatest common factor.
Explanation: This question tests adding, subtracting, factoring, and expanding linear expressions with rational coefficients using properties of operations. Operations: add by combining like terms (3x+2x=5x, coefficients add), subtract by distributing negative then combining (4x-(2x-3)=4x-2x+3=2x+3, negative distributes to all terms), expand using distributive property a(b+c)=ab+ac (multiply each term: 3(2x-5)=6x-15), factor by finding GCF and dividing out (6x+9: GCF=3, so 3(6x/3+9/3)=3(2x+3)). Combining like terms requires same variable (3x and 5x combine, but 2x and 3 don't). For example, to factor 3/5 x + 6/5, find the GCF of 3/5 and 6/5 which is 3/5, then 3/5 (x + 2). In this case, the correct factoring is 3/5 (x + 2). A common error is not identifying the full GCF for fractions or factoring incompletely, such as using 6/5 instead. Strategy: (1) for adding/subtracting: distribute any negatives first (important for subtraction), identify like terms (same variable part: 3x and 5x are like, 2x and 3 aren't), combine (add/subtract coefficients: 3x+5x=8x), combine constants separately (5-3=2); (2) for expanding: distribute to every term (a(b+c+d)=ab+ac+ad, don't miss any); (3) for factoring: find GCF of all terms, divide each term by GCF, write as GCF(quotients). Common mistakes: distributing negative to only first term -(2x-3)=-2x-3 (wrong, should be -2x+3), combining unlike terms, fraction operations without common denominators, factoring incompletely.
Which expression is equivalent to 5(x−52)+3x after simplifying?
Explanation: This question tests adding, subtracting, factoring, and expanding linear expressions with rational coefficients using properties of operations. Operations: add by combining like terms (3x+2x=5x, coefficients add), subtract by distributing negative then combining (4x-(2x-3)=4x-2x+3=2x+3, negative distributes to all terms), expand using distributive property a(b+c)=ab+ac (multiply each term: 3(2x-5)=6x-15), factor by finding GCF and dividing out (6x+9: GCF=3, so 3(6x/3+9/3)=3(2x+3)). Combining like terms requires same variable (3x and 5x combine, but 2x and 3 don't). For example, 5(x - 2/5) + 3x expands to 5x - 2 + 3x, then combines to 8x - 2. In this case, the correct simplification is 8x - 2. A common error is not distributing the 5 to the -2/5 term fully, such as leaving it as 5x - 2/5 + 3x. Strategy: (1) for adding/subtracting: distribute any negatives first (important for subtraction), identify like terms (same variable part: 3x and 5x are like, 2x and 3 aren't), combine (add/subtract coefficients: 3x+5x=8x), combine constants separately (5-3=2); (2) for expanding: distribute to every term (a(b+c+d)=ab+ac+ad, don't miss any); (3) for factoring: find GCF of all terms, divide each term by GCF, write as GCF(quotients). Common mistakes: distributing negative to only first term -(2x-3)=-2x-3 (wrong, should be -2x+3), combining unlike terms, fraction operations without common denominators, factoring incompletely.
In science class, the temperature change is modeled by (4x+7)−(2x+3). Simplify the difference.
Explanation: This question tests adding, subtracting, factoring, and expanding linear expressions with rational coefficients using properties of operations. Operations: add by combining like terms (3x+2x=5x, coefficients add), subtract by distributing negative then combining (4x-(2x-3)=4x-2x+3=2x+3, negative distributes to all terms), expand using distributive property a(b+c)=ab+ac (multiply each term: 3(2x-5)=6x-15), factor by finding GCF and dividing out (6x+9: GCF=3, so 3(6x/3+9/3)=3(2x+3)). Combining like terms requires same variable (3x and 5x combine, but 2x and 3 don't). For example, (4x+7)-(2x+3) requires distributing the negative: 4x+7-2x-3, then combine like terms: 4x-2x=2x, 7-3=4, giving 2x+4. In this case, the correct simplification is (4x+7)-(2x+3)=2x+4. A common error is not distributing the negative to all terms in the subtracted expression, such as treating it as 4x+7-2x+3 instead of -3. Strategy: (1) for adding/subtracting: distribute any negatives first (important for subtraction), identify like terms (same variable part: 3x and 5x are like, 2x and 3 aren't), combine (add/subtract coefficients: 3x+5x=8x), combine constants separately (5-3=2); (2) for expanding: distribute to every term (a(b+c+d)=ab+ac+ad, don't miss any); (3) for factoring: find GCF of all terms, divide each term by GCF, write as GCF(quotients). Common mistakes: distributing negative to only first term -(2x-3)=-2x-3 (wrong, should be -2x+3), combining unlike terms, fraction operations without common denominators, factoring incompletely.
Expand the expression −3(2x−35) using the distributive property.
Explanation: This question tests adding, subtracting, factoring, and expanding linear expressions with rational coefficients using properties of operations. Operations include adding by combining like terms (3x+2x=5x, coefficients add), subtracting by distributing the negative then combining (4x-(2x-3)=4x-2x+3=2x+3, negative distributes to all terms), expanding using the distributive property a(b+c)=ab+ac (multiply each term: 3(2x-5)=6x-15), and factoring by finding the GCF and dividing out (6x+9: GCF=3, so 3(6x/3+9/3)=3(2x+3)). Combining like terms requires the same variable (3x and 5x combine, but 2x and 3 don't). A specific example is expanding 3(2x-5)=6x-15 by distributing to each term. For this problem, distribute -3 to each term: -32x = -6x, -3(-( rac{5}{3})) = +5, resulting in -6x+5. A common error is incomplete distribution, like missing the second term or mishandling the negative sign with the fraction. Strategy: for expanding, distribute to every term (a(b+c+d)=ab+ac+ad, don't miss any); common mistakes include distributing negative to only the first term -(2x-3)=-2x-3 (wrong, should be -2x+3), or fraction operations without proper multiplication.
In science class, the temperature change is modeled by (4x+7)−(2x+3). Simplify the difference.
Explanation: This question tests adding, subtracting, factoring, and expanding linear expressions with rational coefficients using properties of operations. Operations: add by combining like terms (3x+2x=5x, coefficients add), subtract by distributing negative then combining (4x-(2x-3)=4x-2x+3=2x+3, negative distributes to all terms), expand using distributive property a(b+c)=ab+ac (multiply each term: 3(2x-5)=6x-15), factor by finding GCF and dividing out (6x+9: GCF=3, so 3(6x/3+9/3)=3(2x+3)). Combining like terms requires same variable (3x and 5x combine, but 2x and 3 don't). For example, (4x+7)-(2x+3) requires distributing the negative: 4x+7-2x-3, then combine like terms: 4x-2x=2x, 7-3=4, giving 2x+4. In this case, the correct simplification is (4x+7)-(2x+3)=2x+4. A common error is not distributing the negative to all terms in the subtracted expression, such as treating it as 4x+7-2x+3 instead of -3. Strategy: (1) for adding/subtracting: distribute any negatives first (important for subtraction), identify like terms (same variable part: 3x and 5x are like, 2x and 3 aren't), combine (add/subtract coefficients: 3x+5x=8x), combine constants separately (5-3=2); (2) for expanding: distribute to every term (a(b+c+d)=ab+ac+ad, don't miss any); (3) for factoring: find GCF of all terms, divide each term by GCF, write as GCF(quotients). Common mistakes: distributing negative to only first term -(2x-3)=-2x-3 (wrong, should be -2x+3), combining unlike terms, fraction operations without common denominators, factoring incompletely.
During basketball practice, the coach compares two drills. The number of shots is modeled by (4x+7) and (2x-3). Find and simplify the difference (4x+7)-(2x-3).
Explanation: Distribute the negative across the second expression: 4x + 7 - 2x + 3 (since -(2x-3) = -2x+3). Combine the x-terms: 4x - 2x = 2x. Combine the constants: 7 + 3 = 10. Putting these together gives 2x + 10, matching choice D. Choice B (2x+4) comes from only partially distributing the negative, treating -(2x-3) as -2x-3 instead of -2x+3, which flips the sign of the constant. Choices A (2x-10) and C (6x+4) involve further sign or combination errors in the subtraction.