Factor Linear Expressions

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Factor Linear Expressions · Grade 7 · Worksheet 1

1. A community center is planning a mural project that requires rectangular panels. The total area needed for the mural can be expressed as 42x + 56 square feet, where x represents the length of each panel in feet. The project manager needs to factor this expression to determine the standard width that will work for all panels. What is the factored form of the expression 42x + 56 using the greatest common factor? Answer: 14(3x + 4) Solution: Identify the terms in the expression: 42x and 56 Find the greatest common factor of 42 and 56 List the factors of 42: 1, 2, 3, 6, 7, 14, 21, 42 List the factors of 56: 1, 2, 4, 7, 8, 14, 28, 56 The largest common factor is 14 Divide each term by 14: 42x ÷ 14 = 3x, 56 ÷ 14 = 4 Write the factored…
   Full step-by-step solution

   Step 1: Identify the terms in the expression: 42x and 56 Step 2: Find the greatest common factor of 42 and 56 Step 3: List the factors of 42: 1, 2, 3, 6, 7, 14, 21, 42 Step 4: List the factors of 56: 1, 2, 4, 7, 8, 14, 28, 56 Step 5: The largest common factor is 14 Step 6: Divide each term by 14: 42x ÷ 14 = 3x, 56 ÷ 14 = 4 Step 7: Write the factored form: 14(3x + 4) The factored form is 14(3x + 4).

2. A rectangular garden is being designed with length (8x + 12) feet and width (6x + 9) feet. The landscape architect needs to factor both expressions to determine the greatest common side length for square paving stones that will fit evenly along both dimensions. What is the greatest common factor of the length and width expressions? Answer: 2x + 3 Solution: Write down the expressions for length and width. Length = 8x + 12 Width = 6x + 9 Factor each expression separately. First, factor 8x + 12: The greatest common factor of 8x and 12 is 4.
   Full step-by-step solution

   Let's find the greatest common factor of the length and width expressions step by step. --- **Step 1: Write down the expressions for length and width.** Length = 8x + 12 Width = 6x + 9 --- **Step 2: Factor each expression separately.** First, factor 8x + 12: The greatest common factor of 8x and 12 is 4. 8x + 12 = 4(2x) + 4(3) = 4(2x + 3) So, Length = 4(2x + 3) --- Second, factor 6x + 9: The greatest common factor of 6x and 9 is 3. 6x + 9 = 3(2x) + 3(3) = 3(2x + 3) So, Width = 3(2x + 3) --- **Step 3: Compare the factored forms.** Length = 4(2x + 3) Width = 3(2x + 3) --- **Step 4: Identify the common factor.** Both expressions contain the factor (2x + 3). The numeric parts 4 and 3 have no common factor other than 1. Thus, the greatest common factor of the two expressions is (2x + 3). --- **Final Answer:** 2x + 3

3. Factor: 150x + 225 = ? Answer: 75(2x + 3) Solution: Identify the coefficients: 150 and 225. Find the GCF of 150 and 225. - Factors of 150: 1, 2, 3, 5, 6, 10, 15, 25, 30, 50, 75, 150 - Factors of 225: 1, 3, 5, 9, 15, 25, 45, 75, 225 - The greatest common factor is 75.
   Full step-by-step solution

   Step 1: Identify the coefficients: 150 and 225. Step 2: Find the GCF of 150 and 225. - Factors of 150: 1, 2, 3, 5, 6, 10, 15, 25, 30, 50, 75, 150 - Factors of 225: 1, 3, 5, 9, 15, 25, 45, 75, 225 - The greatest common factor is 75. Step 3: Divide each term by 75. - 150x ÷ 75 = 2x - 225 ÷ 75 = 3 Step 4: Write the factored form: 75(2x + 3). The answer is 75(2x + 3).

4. Noah is designing a rectangular mosaic for a community center wall. The mosaic's length is represented by the expression 36x + 66 inches, and its width is represented by the expression 48x + 72 inches. Noah wants to divide the entire mosaic into identical square tiles by drawing grid lines, using the largest possible square tiles so that no tile needs to be cut. What is the side length, in inches, of each square tile? Answer: 6 Solution: The side length of the square tiles must be a common factor of both the length and width expressions. Factor the length expression: 36x + 66. Find the GCF of the coefficients 36 and 66.
   Full step-by-step solution

   Step 1: The side length of the square tiles must be a common factor of both the length and width expressions. Step 2: Factor the length expression: 36x + 66. Find the GCF of the coefficients 36 and 66. Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36 Factors of 66: 1, 2, 3, 6, 11, 22, 33, 66 Common factors: 1, 2, 3, 6 Greatest common factor: 6 So, 36x + 66 = 6(6x + 11) Step 3: Factor the width expression: 48x + 72. Find the GCF of the coefficients 48 and 72. Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48 Factors of 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72 Common factors: 1, 2, 3, 4, 6, 8, 12, 24 Greatest common factor: 24 So, 48x + 72 = 24(2x + 3) Step 4: The side length of the largest square tiles must be the greatest common factor of both entire expressions. This is found by taking the GCF of the numerical factors that are common to both factored forms. The length is 6(6x + 11) and the width is 24(2x + 3). The common numerical factor is the GCF of 6 and 24, which is 6. Step 5: Verify that 6 divides both expressions evenly. (36x + 66) / 6 = 6x + 11, which is a whole number expression. (48x + 72) / 6 = 8x + 12, which is a whole number expression. Therefore, the largest possible side length for the square tiles is 6 inches. The answer is 6.

5. Olivia is planning a large community garden and needs to order fencing. The total length of fencing required, in feet, can be expressed as 105x + 75, where x represents the number of rectangular garden plots. To calculate the cost per plot, she needs to factor this expression by its greatest common factor. What is the factored form of the expression 105x + 75? Answer: 15(7x + 5) Solution: Identify the coefficients in the expression 105x + 75. The coefficients are 105 and 75. Find the greatest common factor (GCF) of 105 and 75.
   Full step-by-step solution

   Step 1: Identify the coefficients in the expression 105x + 75. The coefficients are 105 and 75. Step 2: Find the greatest common factor (GCF) of 105 and 75. - Factors of 105: 1, 3, 5, 7, 15, 21, 35, 105 - Factors of 75: 1, 3, 5, 15, 25, 75 - The common factors are 1, 3, 5, 15. The greatest common factor is 15. Step 3: Factor out the GCF from each term: - 105x divided by 15 equals 7x - 75 divided by 15 equals 5 Step 4: Write the factored form using the distributive property: 105x + 75 = 15(7x + 5) The factored form of the expression is 15(7x + 5).

6. Liam is designing a rectangular garden for his school's environmental club. The total area of the garden can be expressed as 24x + 18 square feet, where x represents the length of one section. To prepare the soil, he needs to find the greatest common factor of the terms in this expression. What is the largest number that divides evenly into both 24x and 18? Answer: 6 Solution: Identify the two numbers we are working with. We have 24x and 18. The "largest number that divides evenly into both" means the greatest common factor (GCF) of 24 and 18, since x is a variable and 18 has no x.
   Full step-by-step solution

   Let's find the largest number that divides evenly into both 24x and 18. Step 1: Identify the two numbers we are working with. We have 24x and 18. The "largest number that divides evenly into both" means the greatest common factor (GCF) of 24 and 18, since x is a variable and 18 has no x. Step 2: Find the prime factorization of each number. 24 = 2 × 12 12 = 2 × 6 6 = 2 × 3 So 24 = 2 × 2 × 2 × 3 = 2^3 × 3 18 = 2 × 9 9 = 3 × 3 So 18 = 2 × 3 × 3 = 2 × 3^2 Step 3: Identify the common factors. Common prime factors between 24 and 18: - Both have at least one factor of 2. - Both have at least one factor of 3. Step 4: Multiply the common factors with the smallest exponent from each number. From 24: 2^3, 3^1 From 18: 2^1, 3^2 Smallest power of 2: 2^1 Smallest power of 3: 3^1 Multiply: 2 × 3 = 6 Step 5: Conclusion. The GCF of 24 and 18 is 6. Since 18 has no x, the GCF of 24x and 18 is 6. Answer: 6