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Factor Linear Expressions

Grade 7 · Algebra · Worksheet 3

  1. A community center is planning a mural project and needs to purchase paint supplies. The total cost for the project can be expressed as 42x + 63 dollars, where x represents the number of paint cans needed. The project manager wants to factor this expression to determine the cost per paint can after accounting for fixed expenses. What is the greatest common factor of the terms 42x and 63? Answer: ______________
  2. A community center is planning to create rectangular activity zones for their summer program. The total area needed for all zones can be expressed as 42x + 63 square meters, where x represents the length of each zone in meters. The director needs to factor this expression to determine the standard width that will work for all zones. What is the factored form of the expression 42x + 63 using the greatest common factor? Answer: ______________
  3. Emma is designing a community mural project that requires rectangular panels. The total area needed for all panels can be expressed as 45x + 75 square feet, where x represents the length of each panel in feet. To optimize material usage, she needs to factor this expression using the greatest common factor. What is the factored form of 45x + 75? Answer: ______________
  4. A community center is planning to create identical care packages for senior citizens. They have collected 72 boxes of tea and 54 jars of honey to distribute equally among the packages with no items left over. What is the greatest number of identical care packages they can create? Answer: ______________
  5. A rectangular mural is being designed for a school hallway. The mural's length is represented by the expression 15x + 25 meters and its width by 12x + 20 meters. The artist wants to divide the mural into identical square sections by drawing grid lines. What is the greatest possible side length, in meters, for each square section so that all sections are identical and cover the entire mural without any gaps or overlaps? Answer: ______________
  6. A shipping company needs to package electronics in rectangular boxes. The total volume of a shipment can be expressed as 24000x + 36000y cubic centimeters, where x represents the length and y represents the width of the boxes. Factor this expression completely to find the greatest common factor that represents the height of all boxes. Answer: ______________
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Answer Key & Explanations

Factor Linear Expressions · Grade 7 · Worksheet 3

  1. A community center is planning a mural project and needs to purchase paint supplies. The total cost for the project can be expressed as 42x + 63 dollars, where x represents the number of paint cans needed. The project manager wants to factor this expression to determine the cost per paint can after accounting for fixed expenses. What is the greatest common factor of the terms 42x and 63? Answer: 21 Solution: Identify the coefficients in the expression 42x + 63. The coefficients are 42 and 63.
    Full step-by-step solution

    Step 1: Identify the coefficients in the expression 42x + 63. The coefficients are 42 and 63. Step 2: Find the factors of 42: 1, 2, 3, 6, 7, 14, 21, 42 Step 3: Find the factors of 63: 1, 3, 7, 9, 21, 63 Step 4: Identify the common factors: 1, 3, 7, 21 Step 5: The greatest common factor is 21. The answer is 21.

  2. A community center is planning to create rectangular activity zones for their summer program. The total area needed for all zones can be expressed as 42x + 63 square meters, where x represents the length of each zone in meters. The director needs to factor this expression to determine the standard width that will work for all zones. What is the factored form of the expression 42x + 63 using the greatest common factor? Answer: 21(2x + 3) Solution: Identify the terms in the expression: 42x and 63 Find the greatest common factor of 42 and 63 The factors of 42 are: 1, 2, 3, 6, 7, 14, 21, 42 The factors of 63 are: 1, 3, 7, 9, 21, 63 The greatest common factor is 21 Divide each term by 21: 42x ÷ 21 = 2x, 63 ÷ 21 = 3 Write the factored form:…
    Full step-by-step solution

    Step 1: Identify the terms in the expression: 42x and 63 Step 2: Find the greatest common factor of 42 and 63 Step 3: The factors of 42 are: 1, 2, 3, 6, 7, 14, 21, 42 Step 4: The factors of 63 are: 1, 3, 7, 9, 21, 63 Step 5: The greatest common factor is 21 Step 6: Divide each term by 21: 42x ÷ 21 = 2x, 63 ÷ 21 = 3 Step 7: Write the factored form: 21(2x + 3) The factored form is 21(2x + 3).

  3. Emma is designing a community mural project that requires rectangular panels. The total area needed for all panels can be expressed as 45x + 75 square feet, where x represents the length of each panel in feet. To optimize material usage, she needs to factor this expression using the greatest common factor. What is the factored form of 45x + 75? Answer: 15(3x + 5) Solution: Identify the terms in the expression: 45x and 75 Find the greatest common factor of 45 and 75 The factors of 45 are: 1, 3, 5, 9, 15, 45 The factors of 75 are: 1, 3, 5, 15, 25, 75 The greatest common factor is 15 Divide each term by 15: 45x ÷ 15 = 3x, 75 ÷ 15 = 5 Write the factored form: 15(3x +…
    Full step-by-step solution

    Step 1: Identify the terms in the expression: 45x and 75 Step 2: Find the greatest common factor of 45 and 75 Step 3: The factors of 45 are: 1, 3, 5, 9, 15, 45 Step 4: The factors of 75 are: 1, 3, 5, 15, 25, 75 Step 5: The greatest common factor is 15 Step 6: Divide each term by 15: 45x ÷ 15 = 3x, 75 ÷ 15 = 5 Step 7: Write the factored form: 15(3x + 5) The factored form is 15(3x + 5).

  4. A community center is planning to create identical care packages for senior citizens. They have collected 72 boxes of tea and 54 jars of honey to distribute equally among the packages with no items left over. What is the greatest number of identical care packages they can create? Answer: 18 Solution: Identify the two quantities: 72 boxes of tea and 54 jars of honey Find the factors of 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72 Find the factors of 54: 1, 2, 3, 6, 9, 18, 27, 54 Identify the common factors: 1, 2, 3, 6, 9, 18 The greatest common factor is 18 Check: 72 ÷ 18 = 4 boxes of tea per…
    Full step-by-step solution

    Step 1: Identify the two quantities: 72 boxes of tea and 54 jars of honey Step 2: Find the factors of 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72 Step 3: Find the factors of 54: 1, 2, 3, 6, 9, 18, 27, 54 Step 4: Identify the common factors: 1, 2, 3, 6, 9, 18 Step 5: The greatest common factor is 18 Step 6: Check: 72 ÷ 18 = 4 boxes of tea per package, 54 ÷ 18 = 3 jars of honey per package The answer is 18.

  5. A rectangular mural is being designed for a school hallway. The mural's length is represented by the expression 15x + 25 meters and its width by 12x + 20 meters. The artist wants to divide the mural into identical square sections by drawing grid lines. What is the greatest possible side length, in meters, for each square section so that all sections are identical and cover the entire mural without any gaps or overlaps? Answer: 5 Solution: Identify that the squares must fit perfectly along both dimensions, so the side length must be a common factor of both the length and width expressions. Factor the length expression: 15x + 25 = 5(3x + 5) Factor the width expression: 12x + 20 = 4(3x + 5) Identify the greatest common factor of…
    Full step-by-step solution

    Step 1: Identify that the squares must fit perfectly along both dimensions, so the side length must be a common factor of both the length and width expressions. Step 2: Factor the length expression: 15x + 25 = 5(3x + 5) Step 3: Factor the width expression: 12x + 20 = 4(3x + 5) Step 4: Identify the greatest common factor of both expressions. Both expressions contain the factor (3x + 5), and the coefficients have GCF of 5 and 4, whose GCF is 1. Step 5: Therefore, the GCF of 15x + 25 and 12x + 20 is (3x + 5). Step 6: The greatest possible side length for the square sections is the coefficient of the GCF, which is the number that multiplies (3x + 5) in both factored forms. Since 5(3x + 5) and 4(3x + 5) both contain (3x + 5), the numerical GCF is 5. The answer is 5.

  6. A shipping company needs to package electronics in rectangular boxes. The total volume of a shipment can be expressed as 24000x + 36000y cubic centimeters, where x represents the length and y represents the width of the boxes. Factor this expression completely to find the greatest common factor that represents the height of all boxes. Answer: 12000 Solution: Identify the coefficients in the expression 24000x + 36000y Find the greatest common factor of 24000 and 36000 Both numbers are divisible by 1000: 24000 ÷ 1000 = 24, 36000 ÷ 1000 = 36 The GCF of 24 and 36 is 12 Multiply back: 1000 × 12 = 12000 Factor out 12000 from the original expression:…
    Full step-by-step solution

    Step 1: Identify the coefficients in the expression 24000x + 36000y Step 2: Find the greatest common factor of 24000 and 36000 Step 3: Both numbers are divisible by 1000: 24000 ÷ 1000 = 24, 36000 ÷ 1000 = 36 Step 4: The GCF of 24 and 36 is 12 Step 5: Multiply back: 1000 × 12 = 12000 Step 6: Factor out 12000 from the original expression: 12000(2x + 3y) Step 7: The GCF 12000 represents the height of the boxes The answer is 12000.