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Expand Linear Expressions

Grade 7 · Algebra · Worksheet 3

  1. A rectangular community garden is drawn on a coordinate grid with corners at (0, 0), (25, 0), (25, 15), and (0, 15). The garden is divided into two rectangular sections by a vertical fence. The left section has a width of 10 units, and the right section has a width of 15 units. Using the distributive property, write an expanded expression for the total area of the garden, and then calculate the total area. Answer: ______________
  2. A rectangular community center is drawn on a coordinate grid. The base of the building has corners at (0, 0), (30, 0), (30, 20), and (0, 20). Liam wants to add a rectangular annex to the right side of the building. The annex has the same height as the original building and a width of 15 units. Write an expanded expression using the distributive property for the total area of the building and annex combined, then calculate that total area. Answer: ______________
  3. Mason is building a rectangular treehouse platform. The platform's length is 12 feet more than twice its width. He decides to add a safety railing that extends 3 feet beyond the platform on all sides. If the original width of the platform is w feet, write a simplified expression in expanded form for the total area of the platform plus the railing extension. Answer: ______________
  4. Mere is helping to organize the school's charity fun run. Each participant pays a registration fee of $15. Additionally, the school receives a fixed donation of $250 from a local sponsor, regardless of the number of participants. If p represents the number of participants, write a simplified expression using the distributive property to show the total amount of money the school receives from the fun run. Answer: ______________
  5. 4(3x - 7) + 2(5 - 2x) = ? Answer: ______________
  6. Hana is helping to organize a school art exhibition. She has a rectangular display board. The length of the board is 10 centimeters more than 4 times its width. She decides to add a decorative frame around the entire board that increases both the length and the width by 6 centimeters. If the original width of the board is represented by w centimeters, write a simplified expression in expanded form for the total area of the display board including the decorative frame. Answer: ______________
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Answer Key & Explanations

Expand Linear Expressions · Grade 7 · Worksheet 3

  1. A rectangular community garden is drawn on a coordinate grid with corners at (0, 0), (25, 0), (25, 15), and (0, 15). The garden is divided into two rectangular sections by a vertical fence. The left section has a width of 10 units, and the right section has a width of 15 units. Using the distributive property, write an expanded expression for the total area of the garden, and then calculate the total area. Answer: 375 Solution: Identify the dimensions. The garden has a height of 15 units (from y=0 to y=15). The left section has a width of 10 units, and the right section has a width of 15 units.
    Full step-by-step solution

    Step 1: Identify the dimensions. The garden has a height of 15 units (from y=0 to y=15). The left section has a width of 10 units, and the right section has a width of 15 units. The total width is 10 + 15 = 25 units. Step 2: Write the total area using the distributive property. Total area = height * (left width + right width) = 15 * (10 + 15). Step 3: Expand using the distributive property: 15 * 10 + 15 * 15 = 150 + 225. Step 4: Calculate the sum: 150 + 225 = 375. The total area of the garden is 375 square units.

  2. A rectangular community center is drawn on a coordinate grid. The base of the building has corners at (0, 0), (30, 0), (30, 20), and (0, 20). Liam wants to add a rectangular annex to the right side of the building. The annex has the same height as the original building and a width of 15 units. Write an expanded expression using the distributive property for the total area of the building and annex combined, then calculate that total area. Answer: 900 Solution: Find the dimensions of the original building. Original width = 30 units (from x=0 to x=30) Height = 20 units (from y=0 to y=20) Area of original = 30 × 20 = 600 square units Find the dimensions of the annex.
    Full step-by-step solution

    Step 1: Find the dimensions of the original building. Original width = 30 units (from x=0 to x=30) Height = 20 units (from y=0 to y=20) Area of original = 30 × 20 = 600 square units Step 2: Find the dimensions of the annex. Annex width = 15 units Annex height = 20 units (same as original) Area of annex = 15 × 20 = 300 square units Step 3: Write the total area using the distributive property. Total area = height × (original width + annex width) Total area = 20 × (30 + 15) Expanded using distributive property: 20 × 30 + 20 × 15 Step 4: Calculate each part. 20 × 30 = 600 20 × 15 = 300 600 + 300 = 900 Step 5: The total area of the building and annex combined is 900 square units. Answer: 900

  3. Mason is building a rectangular treehouse platform. The platform's length is 12 feet more than twice its width. He decides to add a safety railing that extends 3 feet beyond the platform on all sides. If the original width of the platform is w feet, write a simplified expression in expanded form for the total area of the platform plus the railing extension. Answer: 2w^2 + 24w + 54 Solution: Original width = w feet. Original length = 2w + 12 feet (12 more than twice the width). The railing adds 3 feet to each side, so it adds 3 + 3 = 6 feet to both the length and the width.
    Full step-by-step solution

    Step 1: Original width = w feet. Step 2: Original length = 2w + 12 feet (12 more than twice the width). Step 3: The railing adds 3 feet to each side, so it adds 3 + 3 = 6 feet to both the length and the width. Step 4: New width = w + 6 feet. Step 5: New length = (2w + 12) + 6 = 2w + 18 feet. Step 6: Total area = (new length) × (new width) = (2w + 18)(w + 6). Step 7: Expand using the distributive property: (2w)(w) + (2w)(6) + (18)(w) + (18)(6) = 2w^2 + 12w + 18w + 108. Step 8: Combine like terms: 2w^2 + (12w + 18w) + 108 = 2w^2 + 30w + 108. The answer is 2w^2 + 30w + 108.

  4. Mere is helping to organize the school's charity fun run. Each participant pays a registration fee of $15. Additionally, the school receives a fixed donation of $250 from a local sponsor, regardless of the number of participants. If p represents the number of participants, write a simplified expression using the distributive property to show the total amount of money the school receives from the fun run. Answer: 15p + 250 Solution: The total money is the registration fee per participant times the number of participants, plus the fixed donation. Write an expression: 15p + 250. This expression is already simplified, showing the total as 15p + 250.
    Full step-by-step solution

    Step 1: The total money is the registration fee per participant times the number of participants, plus the fixed donation. Step 2: Write an expression: 15p + 250. Step 3: This expression is already simplified, showing the total as 15p + 250. The answer is 15p + 250.

  5. 4(3x - 7) + 2(5 - 2x) = ? Answer: 8x - 18 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(5 - 2x) = 2 × 5 + 2 × (-2x) = 10 - 4x Combine all terms: 12x - 28 + 10 - 4x Combine like terms for x: 12x - 4x = 8x Combine constant terms: -28 + 10 = -18…
    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(5 - 2x) = 2 × 5 + 2 × (-2x) = 10 - 4x Step 3: Combine all terms: 12x - 28 + 10 - 4x Step 4: Combine like terms for x: 12x - 4x = 8x Step 5: Combine constant terms: -28 + 10 = -18 Step 6: Final simplified expression: 8x - 18

  6. Hana is helping to organize a school art exhibition. She has a rectangular display board. The length of the board is 10 centimeters more than 4 times its width. She decides to add a decorative frame around the entire board that increases both the length and the width by 6 centimeters. If the original width of the board is represented by w centimeters, write a simplified expression in expanded form for the total area of the display board including the decorative frame. Answer: 4w^2 + 46w + 120 Solution: Original width = w centimeters. Original length = 4w + 10 centimeters (10 more than 4 times the width).
    Full step-by-step solution

    Step 1: Original width = w centimeters. Step 2: Original length = 4w + 10 centimeters (10 more than 4 times the width). Step 3: The decorative frame adds 6 centimeters to each side, so it adds 6 + 6 = 12 centimeters to both the length and the width. Step 4: New width = w + 12 centimeters. Step 5: New length = (4w + 10) + 12 = 4w + 22 centimeters. Step 6: Total area = (new length) * (new width) = (4w + 22)(w + 12). Step 7: Expand using the distributive property: (4w)(w) + (4w)(12) + (22)(w) + (22)(12) = 4w^2 + 48w + 22w + 264. Step 8: Combine like terms: 4w^2 + (48w + 22w) + 264 = 4w^2 + 70w + 264. The answer is 4w^2 + 70w + 264.