Factor Linear Expressions

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Factor Linear Expressions · Grade 7 · Worksheet 3

1. Isabella is helping her school organize a book donation drive. The total number of books collected can be expressed as 144x + 192, where x represents the number of classrooms that participated. The school wants to package the books into identical boxes for distribution, with each box containing the same number of books from each classroom. What is the greatest common factor of the terms in the expression 144x + 192? Answer: 48 Solution: Identify the coefficients in the expression 144x + 192. They are 144 and 192. Find the prime factorization of 144: 144 = 2 x 72 = 2 x 2 x 36 = 2 x 2 x 2 x 18 = 2 x 2 x 2 x 2 x 9 = 2 x 2 x 2 x 2 x 3 x 3 = 2^4 x 3^2.
   Full step-by-step solution

   Step 1: Identify the coefficients in the expression 144x + 192. They are 144 and 192. Step 2: Find the prime factorization of 144: 144 = 2 x 72 = 2 x 2 x 36 = 2 x 2 x 2 x 18 = 2 x 2 x 2 x 2 x 9 = 2 x 2 x 2 x 2 x 3 x 3 = 2^4 x 3^2. Step 3: Find the prime factorization of 192: 192 = 2 x 96 = 2 x 2 x 48 = 2 x 2 x 2 x 24 = 2 x 2 x 2 x 2 x 12 = 2 x 2 x 2 x 2 x 2 x 6 = 2 x 2 x 2 x 2 x 2 x 2 x 3 = 2^6 x 3. Step 4: Identify the common prime factors. Both have 2 and 3. The smallest power of 2 common to both is 2^4 = 16. The smallest power of 3 common to both is 3^1 = 3. Step 5: Multiply these common factors: 16 x 3 = 48. Step 6: Check: 144 divided by 48 equals 3, and 192 divided by 48 equals 4. So 48 divides both evenly. The greatest common factor of 144x and 192 is 48.

2. 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.

3. A construction company is ordering materials for a large project. They need to calculate the total area of flooring required, which is expressed as 24000x + 36000 square feet, where x represents the length of one section in feet. Factor this expression completely to find the greatest common factor that represents the width of the flooring sections. Answer: 12000 Solution: Identify the coefficients in the expression 24000x + 36000 Find the greatest common factor of 24000 and 36000 Both numbers are divisible by 1000: 24000 ÷ 1000 = 24, 36000 ÷ 1000 = 36 Find the GCF of 24 and 36: 24 = 2 × 2 × 2 × 3, 36 = 2 × 2 × 3 × 3 Common factors are 2, 2, and 3, so GCF = 2 × 2…
   Full step-by-step solution

   Step 1: Identify the coefficients in the expression 24000x + 36000 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: Find the GCF of 24 and 36: 24 = 2 × 2 × 2 × 3, 36 = 2 × 2 × 3 × 3 Step 5: Common factors are 2, 2, and 3, so GCF = 2 × 2 × 3 = 12 Step 6: Multiply back by 1000: 12 × 1000 = 12000 Step 7: Factor out 12000 from the original expression: 12000(2x + 3) Step 8: The greatest common factor is 12000, which represents the width of the flooring sections.

4. Factor the expression: 18x + 24y = ? Answer: 6(3x + 4y) Solution: Identify the coefficients of the terms. The coefficients are 18 (from 18x) and 24 (from 24y). Find the greatest common factor (GCF) of 18 and 24.
   Full step-by-step solution

   Let's factor the expression 18x + 24y step by step. Step 1: Identify the coefficients of the terms. The coefficients are 18 (from 18x) and 24 (from 24y). Step 2: Find the greatest common factor (GCF) of 18 and 24. List the factors: 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. Step 3: Factor out the GCF (6) from each term. 18x divided by 6 is 3x. 24y divided by 6 is 4y. Step 4: Write the expression as the GCF multiplied by the remaining sum. 18x + 24y = 6 * (3x) + 6 * (4y) = 6(3x + 4y) Final answer: 6(3x + 4y)

5. Aroha is helping to organize a large community sports day. The total cost of renting the sports field and equipment can be expressed as 108x + 144 dollars, where x represents the number of hours the field is rented. To apply for a grant, Aroha needs to factor this expression using the greatest common factor. What is the factored form of 108x + 144? Answer: 36(3x + 4) Solution: Identify the coefficients in the expression 108x + 144. They are 108 and 144. Find the greatest common factor (GCF) of 108 and 144.
   Full step-by-step solution

   Step 1: Identify the coefficients in the expression 108x + 144. They are 108 and 144. Step 2: Find the greatest common factor (GCF) of 108 and 144. - Factors of 108: 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 108 - Factors of 144: 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72, 144 - Common factors: 1, 2, 3, 4, 6, 9, 12, 18, 36 - The greatest common factor is 36. Step 3: Factor out the GCF from each term. - 108x divided by 36 equals 3x - 144 divided by 36 equals 4 Step 4: Write the factored form: 36(3x + 4). The factored form of 108x + 144 is 36(3x + 4).

6. A rectangular mosaic is being created using square tiles. The mosaic's length is represented by the expression 18x + 24 centimeters, and its width is represented by 12x + 30 centimeters. The artist wants to use the largest possible square tiles that will fit evenly along both dimensions without cutting any tiles. What is the side length, in centimeters, of these largest square tiles? Answer: 6 Solution: Identify the numerical coefficients from both expressions. From the length (18x + 24), the coefficients are 18 and 24. From the width (12x + 30), the coefficients are 12 and 30.
   Full step-by-step solution

   Step 1: Identify the numerical coefficients from both expressions. From the length (18x + 24), the coefficients are 18 and 24. From the width (12x + 30), the coefficients are 12 and 30. Step 2: Find the greatest common factor (GCF) of all four numbers: 18, 24, 12, and 30. Step 3: Find the prime factors: 18 = 2 × 3 × 3 24 = 2 × 2 × 2 × 3 12 = 2 × 2 × 3 30 = 2 × 3 × 5 Step 4: Identify the common prime factors: All numbers share one 2 and one 3. Step 5: Multiply the common prime factors: 2 × 3 = 6 Step 6: The GCF of 18, 24, 12, and 30 is 6. Step 7: Therefore, the largest square tiles that will fit evenly along both dimensions have a side length of 6 centimeters.

7. Factor the expression: 56x + 84y = ? Answer: 28(2x + 3y) Solution: Identify the coefficients: 56 and 84 Find the greatest common factor (GCF) of 56 and 84 - Factors of 56: 1, 2, 4, 7, 8, 14, 28, 56 - Factors of 84: 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84 - The GCF is 28 - 56x ÷ 28 = 2x - 84y ÷ 28 = 3y - 56x + 84y = 28(2x + 3y) The answer is 28(2x + 3y).
   Full step-by-step solution

   Step 1: Identify the coefficients: 56 and 84 Step 2: Find the greatest common factor (GCF) of 56 and 84 - Factors of 56: 1, 2, 4, 7, 8, 14, 28, 56 - Factors of 84: 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84 - The GCF is 28 Step 3: Divide each term by the GCF - 56x ÷ 28 = 2x - 84y ÷ 28 = 3y Step 4: Write the factored expression - 56x + 84y = 28(2x + 3y) The answer is 28(2x + 3y).