Factor Linear Expressions

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Factor Linear Expressions · Grade 7 · Worksheet 2

1. A community center is planning a mural project that requires rectangular panels. The total area needed for the panels 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 The factors of 42 are: 1, 2, 3, 6, 7, 14, 21, 42 The factors of 56 are: 1, 2, 4, 7, 8, 14, 28, 56 The common factors are: 1, 2, 7, 14 The greatest common factor is 14 Divide each term by 14: 42x ÷ 14 =…
   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: The factors of 42 are: 1, 2, 3, 6, 7, 14, 21, 42 Step 4: The factors of 56 are: 1, 2, 4, 7, 8, 14, 28, 56 Step 5: The common factors are: 1, 2, 7, 14 Step 6: The greatest common factor is 14 Step 7: Divide each term by 14: 42x ÷ 14 = 3x, 56 ÷ 14 = 4 Step 8: Write the factored form: 14(3x + 4) The factored form is 14(3x + 4).

2. Factor the expression: 42x + 56y = ? Answer: 14(3x + 4y) Solution: Identify the coefficients 42 and 56 Find the greatest common factor of 42 and 56 Factors of 42: 1, 2, 3, 6, 7, 14, 21, 42 Factors of 56: 1, 2, 4, 7, 8, 14, 28, 56 The greatest common factor is 14 Divide each term by 14: 42x ÷ 14 = 3x, 56y ÷ 14 = 4y Write the factored expression: 14(3x + 4y) The…
   Full step-by-step solution

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

3. A rectangular garden is designed with a length of 24 meters and a width of 18 meters. The gardener wants to divide the entire garden into identical square plots for different vegetables. What is the largest possible side length (in meters) for each square plot so that no space is wasted? Answer: 6 Solution: We are given a rectangular garden with length 24 m and width 18 m. We want to divide it into identical square plots without wasting space.
   Full step-by-step solution

   We are given a rectangular garden with length 24 m and width 18 m. We want to divide it into identical square plots without wasting space. That means the side length of each square must divide both the length and the width evenly. Step 1: Understand the requirement Let the side length of each square be \( s \) meters. Then \( s \) must divide 24 and \( s \) must divide 18. Also, we want the largest possible \( s \). Step 2: Identify the mathematical concept The largest number that divides both 24 and 18 is their Greatest Common Divisor (GCD). Step 3: Find GCD of 24 and 18 List factors of 24: 1, 2, 3, 4, 6, 8, 12, 24 List factors of 18: 1, 2, 3, 6, 9, 18 Common factors: 1, 2, 3, 6 Largest common factor: 6 Step 4: Verify If square side = 6 m: Along length 24 m: 24/6 = 4 squares Along width 18 m: 18/6 = 3 squares Total squares = 4 × 3 = 12 squares Area of 12 squares = 12 × (6 × 6) = 12 × 36 = 432 m² Area of garden = 24 × 18 = 432 m² Matches — no space wasted. Step 5: Conclusion The largest possible side length for each square plot is 6 meters.

4. Factor: 48x + 72 = ? Answer: 24(2x + 3) Solution: Identify the coefficients: 48 and 72. Find the GCF of 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 The GCF is 24.
   Full step-by-step solution

   Step 1: Identify the coefficients: 48 and 72. Step 2: Find the GCF of 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 The GCF is 24. Step 3: Divide each term by the GCF. 48x ÷ 24 = 2x 72 ÷ 24 = 3 Step 4: Write the factored form: 24(2x + 3) The answer is 24(2x + 3).

5. 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

6. Factor the expression: 18x + 24 = ? Answer: 6(3x + 4) Solution: To factor the expression 18x + 24, we need to find the greatest common factor (GCF) of the two terms, 18x and 24. Find the GCF of the numerical coefficients. The coefficients are 18 and 24.
   Full step-by-step solution

   To factor the expression 18x + 24, we need to find the greatest common factor (GCF) of the two terms, 18x and 24. Step 1: Find the GCF of the numerical coefficients. The coefficients are 18 and 24. List the factors of each number: - Factors of 18: 1, 2, 3, 6, 9, 18 - Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24 The largest common factor is 6. So, the GCF is 6. Step 2: Write each term of the expression as a product of the GCF and another factor. - The first term is 18x. We can write this as 6 * (3x) because 6 * 3x = 18x. - The second term is 24. We can write this as 6 * (4) because 6 * 4 = 24. Step 3: Factor the GCF out of the expression. Now we rewrite the original expression using the products from Step 2: 18x + 24 = (6 * 3x) + (6 * 4) Step 4: Apply the distributive property in reverse. The distributive property states that a(b + c) = ab + ac. We are doing the reverse: ab + ac = a(b + c). Here, the common factor 'a' is 6. So, (6 * 3x) + (6 * 4) = 6 * (3x + 4) Therefore, the factored form of 18x + 24 is 6(3x + 4). We can check our work by distributing the 6 back into the parentheses: 6 * (3x + 4) = (6 * 3x) + (6 * 4) = 18x + 24. This matches the original expression, confirming our answer is correct.