Middle School Math Quiz: Add Subtract Factor Linear Expressions
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Add Subtract Factor Linear ExpressionsQuestion 1 of 20

Expand the expression −3(2x−53)-3\left(2x-\frac{5}{3}\right) using the distributive property.

6x−56x-5
−6x+5-6x+5
−3x+53-3x+\frac{5}{3}
−6x−5-6x-5
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Middle School Math Quiz

Middle School Math Quiz: Add Subtract Factor Linear Expressions

Practice Add Subtract Factor Linear Expressions in Middle School Math with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

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 Middle School Math.

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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.

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Question 1

Expand the expression −3(2x−53)-3\left(2x-\frac{5}{3}\right) using the distributive property.

  1. 6x−56x-5
  2. −6x+5-6x+5 (correct answer)
  3. −3x+53-3x+\frac{5}{3}
  4. −6x−5-6x-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 expanding 3(2x-5)=6x-15 by distributing to each term. For this problem, distribute -3 to each term: -32x = -6x, -3(-(53\frac{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.

Question 2

Expand 34(8x−6)\frac{3}{4}\left(8x-6\right) using the distributive property.

  1. 34x−6\frac{3}{4}x-6
  2. 8x−928x-\frac{9}{2}
  3. 6x−926x-\frac{9}{2} (correct answer)
  4. 6x−1846x-\frac{18}{4}
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 (34\frac{3}{4}) to each term: (34\frac{3}{4} imes 8x = 6x), (\frac{3}{4} imes (-6) = -\frac{18}{4} = -\frac{9}{2}), giving (6x - 92\frac{9}{2}). A common error is incomplete distribution, like missing the second term or improper fraction multiplication. Strategy: for expanding, distribute to every term (a(b+c+d)=ab+ac+ad, don't miss any); common mistakes include fraction operations without proper multiplication or simplification.

Question 3

When factoring 21a−14b+35c21a - 14b + 35c, a student writes 7(3a−2b+5c)7(3a - 2b + 5c). What should the student do to verify this factorization is correct?

  1. Check that 77 divides evenly into 2121, 1414, and 3535 without remainders
  2. Confirm that no larger number than 77 can be factored from all three terms
  3. Use the distributive property to multiply 7(3a−2b+5c)7(3a - 2b + 5c) and compare to the original (correct answer)
  4. Verify that the variables aa, bb, and cc appear in both expressions with identical coefficients
Explanation: To verify any factorization, expand it using the distributive property: 7(3a−2b+5c)=21a−14b+35c7(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.

Question 4

Combine like terms to simplify (2x−34)+(−5x+12)\left(2x-\frac{3}{4}\right)+\left(-5x+\frac{1}{2}\right).​

  1. 3x−143x-\frac{1}{4}
  2. −7x−14-7x-\frac{1}{4}
  3. −3x+54-3x+\frac{5}{4}
  4. −3x−14-3x-\frac{1}{4} (correct answer)
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.

Question 5

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).

  1. 5/6 x + 3 (correct answer)
  2. 1/2 x + 3
  3. 3/9 x + 7
  4. 5/6 x - 3
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.

Question 6

A coach writes an expression for total practice time: 3(23x−5)3\left(\frac{2}{3}x-5\right). Expand the expression using the distributive property.

  1. 2x−52x-5
  2. 2x−152x-15 (correct answer)
  3. 23x−15\frac{2}{3}x-15
  4. 2x+152x+15
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.

Question 7

Simplify the expression 0.5x+2.5−0.3x+1.50.5x+2.5-0.3x+1.5 by combining like terms.​

  1. 0.5x+40.5x+4
  2. 0.2x+40.2x+4 (correct answer)
  3. 0.2x+10.2x+1
  4. 0.8x+40.8x+4
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.

Question 8

Factor the expression 35x+65\frac{3}{5}x+\frac{6}{5} by taking out the greatest common factor.​

  1. 35(x+65)\frac{3}{5}(x+\frac{6}{5})
  2. 310(2x+6)\frac{3}{10}(2x+6)
  3. 65(x+1)\frac{6}{5}(x+1)
  4. 35(x+2)\frac{3}{5}(x+2) (correct answer)
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.

Question 9

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).

  1. 2x-10
  2. 2x+4
  3. 6x+4
  4. 2x+10 (correct answer)
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.

Question 10

Combine like terms to simplify (2x−34)+(−5x+12)\left(2x-\frac{3}{4}\right)+\left(-5x+\frac{1}{2}\right).

  1. −3x−14-3x-\frac{1}{4} (correct answer)
  2. −7x−14-7x-\frac{1}{4}
  3. −3x+54-3x+\frac{5}{4}
  4. 3x−143x-\frac{1}{4}
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.

Question 11

Expand the expression for total cost: −2(3x−72)-2\left(3x-\frac{7}{2}\right).

  1. 6x−76x-7
  2. −6x+7-6x+7 (correct answer)
  3. −6x−7-6x-7
  4. −5x+7-5x+7
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 -2: -23x = -6x, -2(-7/2) = +7, giving -6x + 7, which is choice B. A common error is sign error in distribution, like -2*(-7/2) as -7 instead of +7. 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.

Question 12

A coach says the total practice time can be written as 3(2x−5)3(2x-5). Expand the expression using the distributive property.

  1. 6(x−5)6(x-5)
  2. 6x−56x-5
  3. 2x−152x-15
  4. 6x−156x-15 (correct answer)
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.

Question 13

A student factored the expression 12x+18y−612x + 18y - 6 and wrote 6(2x+3y−1)6(2x + 3y - 1). To verify this is correct, which property should be applied?

  1. The commutative property to rearrange terms before checking the multiplication
  2. The associative property to group terms differently before factoring out the GCF
  3. The distributive property to multiply 66 by each term inside the parentheses (correct answer)
  4. The identity property to confirm that factoring doesn't change the expression's value
Explanation: To verify factoring is correct, use the distributive property to expand 6(2x+3y−1)=12x+18y−66(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.

Question 14

A student writes two expressions for points scored in a game: (34x+5)\left(\frac{3}{4}x+5\right) and (12x−3)\left(\frac{1}{2}x-3\right). Simplify the sum (34x+5)+(12x−3)\left(\frac{3}{4}x+5\right)+\left(\frac{1}{2}x-3\right).​

  1. 54x−8\frac{5}{4}x-8
  2. 78x+2\frac{7}{8}x+2
  3. 14x+2\frac{1}{4}x+2
  4. 54x+2\frac{5}{4}x+2 (correct answer)
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, (3/4 x + 5) + (1/2 x - 3) involves combining like terms: first convert 1/2 to 2/4, so 3/4 x + 2/4 x = 5/4 x, and 5 + (-3) = 2, resulting in 5/4 x + 2. In this case, the correct simplification is (3/4 x + 5) + (1/2 x - 3) = 5/4 x + 2. A common error is forgetting to find a common denominator when adding fractions, such as adding 3/4 + 1/2 directly as 3/4 + 1/2 = 4/6 instead of 5/4. 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.

Question 15

Combine like terms to simplify: 0.6x+2.5−0.3x+1.20.6x+2.5-0.3x+1.2.

  1. 0.2x+3.70.2x+3.7
  2. 0.3x+3.70.3x+3.7 (correct answer)
  3. 0.3x+1.30.3x+1.3
  4. 0.9x+3.70.9x+3.7
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 (3x+5)+(2x-3): combine 3x+2x=5x, 5-3=2, giving 5x+2. For this problem, combine x terms: 0.6x - 0.3x = 0.3x, constants: 2.5 + 1.2 = 3.7, resulting in 0.3x + 3.7. A common error is combining unlike terms, like adding a variable and constant. Strategy: for adding/subtracting, 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 combining unlike terms.

Question 16

Which expression is equivalent to 2(35x−4)+(15x+6)2\left(\frac{3}{5}x-4\right)+\left(\frac{1}{5}x+6\right) after simplifying?

  1. 65x−2\frac{6}{5}x-2
  2. 75x−2\frac{7}{5}x-2 (correct answer)
  3. 75x+14\frac{7}{5}x+14
  4. 710x−2\frac{7}{10}x-2
Explanation: Distributing the 2 across the first parentheses gives 2 times 3/5x, which is 6/5x, minus 2 times 4, which is 8, so the expression becomes 6/5x minus 8, plus 1/5x plus 6, matching the setup for choice B. Combining the x terms, 6/5x plus 1/5x equals 7/5x, and combining the constants, negative 8 plus 6 equals negative 2, giving a final answer of 7/5x minus 2. Choice A, 6/5x minus 2, is missing the 1/5x term, likely from forgetting to add it in after distributing. Choice C, 7/5x plus 14, comes from combining the constants incorrectly instead of adding negative 8 and 6. Choice D, 7/10x minus 2, comes from an error combining the fractional coefficients, treating them as if they needed a common denominator of 10 instead of adding them directly since they already share a denominator of 5.

Question 17

When subtracting 2(3x−5)2(3x - 5) from 4(2x+3)4(2x + 3), what is the coefficient of xx in the simplified result?

  1. 22 (correct answer)
  2. 1414
  3. −2-2
  4. 66
Explanation: Subtracting 2(3x−5)2(3x - 5) from 4(2x+3)4(2x + 3) means: 4(2x+3)−2(3x−5)=8x+12−6x+10=2x+224(2x + 3) - 2(3x - 5) = 8x + 12 - 6x + 10 = 2x + 22. The coefficient of xx is 22. Choice B comes from adding instead of subtracting: 8x+6x=14x8x + 6x = 14x. Choice C results from calculating 6x−8x=−2x6x - 8x = -2x. Choice D comes from incorrectly distributing.

Question 18

Simplify 15xy−25x+10x15xy - 25x + 10x by combining like terms, and then write the complete factorization of the simplified expression.

  1. 5x(3y−5+2)5x(3y - 5 + 2)
  2. 5x(3y−3)5x(3y - 3) (correct answer)
  3. 10x(1.5y−2.5+1)10x(1.5y - 2.5 + 1)
  4. x(15y−15)x(15y - 15)
Explanation: First combine the like terms -25x and +10x to get 15xy - 15x. Then factor out the greatest common factor, which is 5x, giving 5x(3y - 3). Choice A shows factoring before combining -25x and +10x, which skips the required simplification step. Choice C uses 10x as the factor, resulting in non-integer coefficients inside the parentheses, instead of the correct GCF of 5x. Choice D factors out only x, not the full greatest common factor of 5x.

Question 19

Factor the linear expression completely: 9x-12.

  1. 3(3x-4) (correct answer)
  2. -3(3x+4)
  3. 6(3/2 x-2)
  4. 9(x-12)
Explanation: The GCF of 9x and -12 is 3. Dividing each term by 3 gives 9x/3 = 3x and -12/3 = -4, so the factored form is 3(3x - 4), matching choice A. Choice B (-3(3x+4)) uses the wrong sign for the GCF, which flips the signs inside the parentheses incorrectly. Choice C uses 6, which isn't the GCF of 9 and 12, so the x-coefficient becomes a fraction (3/2) instead of a whole number. Choice D doesn't divide -12 by 9 at all: 9(x-12) actually equals 9x - 108, not the original expression.

Question 20

A student writes two expressions for points scored in a game: (34x+5)\left(\frac{3}{4}x+5\right) and (12x−3)\left(\frac{1}{2}x-3\right). Simplify the sum (34x+5)+(12x−3)\left(\frac{3}{4}x+5\right)+\left(\frac{1}{2}x-3\right).

  1. 14x+2\frac{1}{4}x+2
  2. 54x+2\frac{5}{4}x+2 (correct answer)
  3. 78x+2\frac{7}{8}x+2
  4. 54x−8\frac{5}{4}x-8
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, (3/4 x + 5) + (1/2 x - 3) involves combining like terms: first convert 1/2 to 2/4, so 3/4 x + 2/4 x = 5/4 x, and 5 + (-3) = 2, resulting in 5/4 x + 2. In this case, the correct simplification is (3/4 x + 5) + (1/2 x - 3) = 5/4 x + 2. A common error is forgetting to find a common denominator when adding fractions, such as adding 3/4 + 1/2 directly as 3/4 + 1/2 = 4/6 instead of 5/4. 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.