Expand Linear Expressions

lessonbunny.com

Answer Key & Explanations

Expand Linear Expressions · Grade 7 · Worksheet 1

1. A rectangular playground is drawn on a coordinate grid with corners at (0, 0), (18, 0), (18, 12), and (0, 12). Charlotte wants to build a new rectangular sandbox that is attached to the right side of the playground. The sandbox has the same height as the playground and a width of 7 units. Using the distributive property, write an expression for the total area of the playground and sandbox combined, then calculate the total area. Answer: 300 Solution: Find the dimensions of the original playground. The width is the difference in x-coordinates: 18 - 0 = 18 units. The height is the difference in y-coordinates: 12 - 0 = 12 units.
   Full step-by-step solution

   Step 1: Find the dimensions of the original playground. The width is the difference in x-coordinates: 18 - 0 = 18 units. The height is the difference in y-coordinates: 12 - 0 = 12 units. Area of playground = 18 * 12 = 216 square units. Step 2: The sandbox has the same height of 12 units and a width of 7 units. Area of sandbox = 7 * 12 = 84 square units. Step 3: Write the total area using the distributive property. Total area = height * (playground width + sandbox width) = 12 * (18 + 7). Step 4: Use the distributive property: 12 * (18 + 7) = 12 * 18 + 12 * 7 = 216 + 84 = 300. Step 5: The total area is 300 square units.

2. Liam is designing a rectangular garden for his school project. The length of the garden is represented by (3x + 5) feet, and the width is (2x - 1) feet. He needs to calculate the total area of the garden to determine how much soil to buy. What is the expanded expression, in square feet, that represents the area of Liam's garden? Answer: 6x^2 + 7x - 5 Solution: To find the area of a rectangle, we multiply the length by the width. Length = (3x + 5) feet Width = (2x - 1) feet Area = Length × Width Area = (3x + 5) × (2x - 1) We will expand this using the distributive property, often remembered by the acronym FOIL (First, Outer, Inner, Last).
   Full step-by-step solution

   To find the area of a rectangle, we multiply the length by the width. Length = (3x + 5) feet Width = (2x - 1) feet Area = Length × Width Area = (3x + 5) × (2x - 1) We will expand this using the distributive property, often remembered by the acronym FOIL (First, Outer, Inner, Last). Step 1: Multiply the First terms. First: (3x) × (2x) = 6x^2 Step 2: Multiply the Outer terms. Outer: (3x) × (-1) = -3x Step 3: Multiply the Inner terms. Inner: (5) × (2x) = 10x Step 4: Multiply the Last terms. Last: (5) × (-1) = -5 Now, we write down all the products we calculated: Area = 6x^2 + (-3x) + 10x + (-5) Step 5: Combine the like terms (the x terms). The x terms are -3x and +10x. -3x + 10x = 7x Step 6: Write the final simplified expression. Area = 6x^2 + 7x - 5 Therefore, the expanded expression that represents the area of Liam's garden is 6x^2 + 7x - 5 square feet.

3. Olivia is designing a rectangular garden on a coordinate grid. The garden has corners at (0, 0), (17, 0), (17, 9), and (0, 9). She decides to divide the garden into two rectangular sections by adding a fence parallel to the left and right sides, located 7 units from the left edge. Using the distributive property, write an expression for the total area of the garden and then calculate the total area. Answer: 153 Solution: Identify the dimensions of the whole garden. Length (horizontal) = 17 - 0 = 17 units Width (vertical) = 9 - 0 = 9 units Total area = length x width = 17 x 9 Write the total area using the distributive property.
   Full step-by-step solution

   Step 1: Identify the dimensions of the whole garden. Length (horizontal) = 17 - 0 = 17 units Width (vertical) = 9 - 0 = 9 units Total area = length x width = 17 x 9 Step 2: Write the total area using the distributive property. The fence divides the length into two parts: 7 units and 10 units (since 17 = 7 + 10). So, total area = 9 x (7 + 10) Using the distributive property: 9 x 7 + 9 x 10 Step 3: Calculate step by step. 9 x 7 = 63 9 x 10 = 90 63 + 90 = 153 Step 4: Verify by direct multiplication. 17 x 9 = 153 The total area of the garden is 153 square units.

4. 3(4x - 7) + 2(5x + 3) = ? Answer: 22x - 15 Solution: Distribute 3 to both terms inside the first parentheses: 3 × 4x = 12x and 3 × (-7) = -21, so we get 12x - 21 Distribute 2 to both terms inside the second parentheses: 2 × 5x = 10x and 2 × 3 = 6, so we get 10x + 6 Combine all terms: (12x - 21) + (10x + 6) Combine like terms: 12x + 10x = 22x and…
   Full step-by-step solution

   Step 1: Distribute 3 to both terms inside the first parentheses: 3 × 4x = 12x and 3 × (-7) = -21, so we get 12x - 21 Step 2: Distribute 2 to both terms inside the second parentheses: 2 × 5x = 10x and 2 × 3 = 6, so we get 10x + 6 Step 3: Combine all terms: (12x - 21) + (10x + 6) Step 4: Combine like terms: 12x + 10x = 22x and -21 + 6 = -15 Step 5: The simplified expression is 22x - 15

5. 4(3x - 7) + 2(5x + 6) = ? Answer: 22x - 16 Solution: Apply distributive property to the first term: 4(3x - 7) = 4 × 3x + 4 × (-7) = 12x - 28 Apply distributive property to the second term: 2(5x + 6) = 2 × 5x + 2 × 6 = 10x + 12 Combine all terms: (12x - 28) + (10x + 12) = 12x + 10x - 28 + 12 Combine like terms: 12x + 10x = 22x and -28 + 12 = -16…
   Full step-by-step solution

   Step 1: Apply distributive property to the first term: 4(3x - 7) = 4 × 3x + 4 × (-7) = 12x - 28 Step 2: Apply distributive property to the second term: 2(5x + 6) = 2 × 5x + 2 × 6 = 10x + 12 Step 3: Combine all terms: (12x - 28) + (10x + 12) = 12x + 10x - 28 + 12 Step 4: Combine like terms: 12x + 10x = 22x and -28 + 12 = -16 Step 5: Final simplified expression: 22x - 16

6. 4(2x - 7) + 3(5 - x) = ? Answer: 5x - 13 Solution: Distribute 4 to both terms inside the first parentheses: 4 × 2x = 8x and 4 × (-7) = -28 Distribute 3 to both terms inside the second parentheses: 3 × 5 = 15 and 3 × (-x) = -3x Write the expanded expression: 8x - 28 + 15 - 3x Combine like terms for x: 8x - 3x = 5x Combine constant terms: -28 + 15…
   Full step-by-step solution

   Step 1: Distribute 4 to both terms inside the first parentheses: 4 × 2x = 8x and 4 × (-7) = -28 Step 2: Distribute 3 to both terms inside the second parentheses: 3 × 5 = 15 and 3 × (-x) = -3x Step 3: Write the expanded expression: 8x - 28 + 15 - 3x Step 4: Combine like terms for x: 8x - 3x = 5x Step 5: Combine constant terms: -28 + 15 = -13 Step 6: Write the simplified expression: 5x - 13 The answer is 5x - 13.

7. Isabella is helping her school's drama club build a rectangular stage for the upcoming play. The original stage has a length of (9x + 12) meters and a width of (7x - 4) meters. The director decides to extend the stage by adding a uniform walkway around all four sides that increases both the length and width by 8 meters. What is the simplified expression for the total area of the expanded stage including the walkway? Answer: 63x^2 + 134x - 48 Solution: Original length = 9x + 12, original width = 7x - 4 Adding 8 meters to all sides means adding 8 to both ends of length and width: new length = (9x + 12) + 8 + 8 = 9x + 28, new width = (7x - 4) + 8 + 8 = 7x + 12 Area = (new length) × (new width) = (9x + 28)(7x + 12) Use distributive property:…
   Full step-by-step solution

   Step 1: Original length = 9x + 12, original width = 7x - 4 Step 2: Adding 8 meters to all sides means adding 8 to both ends of length and width: new length = (9x + 12) + 8 + 8 = 9x + 28, new width = (7x - 4) + 8 + 8 = 7x + 12 Step 3: Area = (new length) × (new width) = (9x + 28)(7x + 12) Step 4: Use distributive property: 9x(7x) + 9x(12) + 28(7x) + 28(12) = 63x^2 + 108x + 196x + 336 Step 5: Combine like terms: 63x^2 + (108x + 196x) + 336 = 63x^2 + 304x + 336 The answer is 63x^2 + 304x + 336.