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Expand Linear Expressions

Grade 7 · Algebra · Worksheet 2

  1. 5(10x - 15) + 3(20 - 5x) = ? Answer: ______________
  2. A rectangular community garden plot has a length that is 5 meters more than three times its width. The gardeners want to expand the plot by adding a uniform pathway around all sides that increases both dimensions by 4 meters. If the original width is w meters, write a simplified expression for the total area of the expanded garden plot including the pathway. Answer: ______________
  3. Aroha is designing a rectangular community garden. The length of the garden is 14 meters more than 9 times its width. To make it wheelchair accessible, a uniform concrete path of 3 meters wide will be added around the entire garden. If the original width of the garden is represented by w meters, write a simplified expression in expanded form for the total area of the garden including the concrete path. Answer: ______________
  4. 7(17x - 22) + 2(37 - 12x) = ? Answer: ______________
  5. A rectangular community garden is being expanded. The original garden has a length of (4x + 7) meters and a width of (3x - 2) meters. The city wants to add a walking path around the entire garden that increases both dimensions by 5 meters. Write a simplified expression for the total area of the expanded garden including the walking path. Answer: ______________
  6. Aroha is organizing a school fundraiser and needs to calculate the total cost for a rectangular banner. The length of the banner is 7 meters more than three times the width. She decides to add a decorative border around the entire banner that increases both the length and the width by 3 meters. If the original width is represented by w meters, write a simplified expression for the total area of the banner including the decorative border in expanded form. Answer: ______________
  7. A rectangular garden is drawn on a coordinate plane with corners at (2, 1), (8, 1), (8, 5), and (2, 5). If the gardener wants to plant flowers along the entire perimeter of the garden, how many units of fencing will she need to enclose the entire garden? Answer: ______________
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Answer Key & Explanations

Expand Linear Expressions · Grade 7 · Worksheet 2

  1. 5(10x - 15) + 3(20 - 5x) = ? Answer: 35x - 15 Solution: Write the expanded expression: 50x - 75 + 60 - 15x. Combine like terms for x: 50x - 15x = 35x. Combine constant terms: -75 + 60 = -15.
    Full step-by-step solution

    Step 1: Apply the distributive property to the first term: 5(10x - 15) = 5 × 10x + 5 × (-15) = 50x - 75. Step 2: Apply the distributive property to the second term: 3(20 - 5x) = 3 × 20 + 3 × (-5x) = 60 - 15x. Step 3: Write the expanded expression: 50x - 75 + 60 - 15x. Step 4: Combine like terms for x: 50x - 15x = 35x. Step 5: Combine constant terms: -75 + 60 = -15. Step 6: Write the simplified expression: 35x - 15. The answer is 35x - 15.

  2. A rectangular community garden plot has a length that is 5 meters more than three times its width. The gardeners want to expand the plot by adding a uniform pathway around all sides that increases both dimensions by 4 meters. If the original width is w meters, write a simplified expression for the total area of the expanded garden plot including the pathway. Answer: 3w^2 + 33w + 56 Solution: Step 1: Original width = w meters Step 2: Original length = 3w + 5 meters (5 more than three times the width) Step 3: After adding 4 meters to both sides, new width = w + 4 + 4 = w + 8 meters Step 4: After adding 4 meters to both sides, new length = (3w + 5) + 4 + 4 = 3w + 13 meters Step 5: Area…
    Full step-by-step solution

    Step 1: Original width = w meters Step 2: Original length = 3w + 5 meters (5 more than three times the width) Step 3: After adding 4 meters to both sides, new width = w + 4 + 4 = w + 8 meters Step 4: After adding 4 meters to both sides, new length = (3w + 5) + 4 + 4 = 3w + 13 meters Step 5: Area of expanded garden = (new length) × (new width) = (3w + 13)(w + 8) Step 6: Expand using distributive property: (3w)(w) + (3w)(8) + (13)(w) + (13)(8) Step 7: Calculate each term: 3w^2 + 24w + 13w + 56 Step 8: Combine like terms: 3w^2 + 37w + 56 Step 9: Verify: 24w + 13w = 37w The simplified expression is 3w^2 + 37w + 56.

  3. Aroha is designing a rectangular community garden. The length of the garden is 14 meters more than 9 times its width. To make it wheelchair accessible, a uniform concrete path of 3 meters wide will be added around the entire garden. If the original width of the garden is represented by w meters, write a simplified expression in expanded form for the total area of the garden including the concrete path. Answer: 9w^2 + 106w + 105 Solution: Original width = w meters. Original length = 9w + 14 meters (14 more than 9 times the width). The path adds 3 meters to each side, so it adds 3 + 3 = 6 meters to both the length and the width.
    Full step-by-step solution

    Step 1: Original width = w meters. Step 2: Original length = 9w + 14 meters (14 more than 9 times the width). Step 3: The path adds 3 meters to each side, so it adds 3 + 3 = 6 meters to both the length and the width. Step 4: New width = w + 6 meters. Step 5: New length = (9w + 14) + 6 = 9w + 20 meters. Step 6: Total area = (new length) x (new width) = (9w + 20)(w + 6). Step 7: Expand using the distributive property: (9w)(w) + (9w)(6) + (20)(w) + (20)(6) = 9w^2 + 54w + 20w + 120. Step 8: Combine like terms: 9w^2 + (54w + 20w) + 120 = 9w^2 + 74w + 120. The answer is 9w^2 + 74w + 120.

  4. 7(17x - 22) + 2(37 - 12x) = ? Answer: 95x - 80 Solution: Write the expanded expression: 119x - 154 + 74 - 24x. Combine like terms for x: 119x - 24x = 95x. Combine constant terms: -154 + 74 = -80.
    Full step-by-step solution

    Step 1: Apply the distributive property to the first term: 7(17x - 22) = 7 × 17x + 7 × (-22) = 119x - 154. Step 2: Apply the distributive property to the second term: 2(37 - 12x) = 2 × 37 + 2 × (-12x) = 74 - 24x. Step 3: Write the expanded expression: 119x - 154 + 74 - 24x. Step 4: Combine like terms for x: 119x - 24x = 95x. Step 5: Combine constant terms: -154 + 74 = -80. Step 6: Write the simplified expression: 95x - 80. The answer is 95x - 80.

  5. A rectangular community garden is being expanded. The original garden has a length of (4x + 7) meters and a width of (3x - 2) meters. The city wants to add a walking path around the entire garden that increases both dimensions by 5 meters. Write a simplified expression for the total area of the expanded garden including the walking path. Answer: 12x^2 + 71x + 24 Solution: Original length = (4x + 7) meters, original width = (3x - 2) meters The walking path adds 5 meters to both dimensions, so: New length = (4x + 7) + 5 = (4x + 12) meters New width = (3x - 2) + 5 = (3x + 3) meters Area of expanded garden = new length × new width Area = (4x + 12)(3x + 3) First: 4x ×…
    Full step-by-step solution

    Step 1: Original length = (4x + 7) meters, original width = (3x - 2) meters Step 2: The walking path adds 5 meters to both dimensions, so: New length = (4x + 7) + 5 = (4x + 12) meters New width = (3x - 2) + 5 = (3x + 3) meters Step 3: Area of expanded garden = new length × new width Area = (4x + 12)(3x + 3) Step 4: Use distributive property (FOIL method): First: 4x × 3x = 12x^2 Outer: 4x × 3 = 12x Inner: 12 × 3x = 36x Last: 12 × 3 = 36 Step 5: Combine like terms: 12x^2 + (12x + 36x) + 36 = 12x^2 + 48x + 36 Step 6: The simplified expression is 12x^2 + 48x + 36 square meters

  6. Aroha is organizing a school fundraiser and needs to calculate the total cost for a rectangular banner. The length of the banner is 7 meters more than three times the width. She decides to add a decorative border around the entire banner that increases both the length and the width by 3 meters. If the original width is represented by w meters, write a simplified expression for the total area of the banner including the decorative border in expanded form. Answer: 3w^2 + 31w + 60 Solution: Original width = w meters Original length = 3w + 7 meters (7 more than three times the width) After adding the 3-meter border, new width = w + 3 + 3 = w + 6 meters After adding the 3-meter border, new length = (3w + 7) + 3 + 3 = 3w + 13 meters Total area = (new length) x (new width) = (3w +…
    Full step-by-step solution

    Step 1: Original width = w meters Step 2: Original length = 3w + 7 meters (7 more than three times the width) Step 3: After adding the 3-meter border, new width = w + 3 + 3 = w + 6 meters Step 4: After adding the 3-meter border, new length = (3w + 7) + 3 + 3 = 3w + 13 meters Step 5: Total area = (new length) x (new width) = (3w + 13)(w + 6) Step 6: Expand using the distributive property: (3w)(w) + (3w)(6) + (13)(w) + (13)(6) = 3w^2 + 18w + 13w + 78 Step 7: Combine like terms: 3w^2 + (18w + 13w) + 78 = 3w^2 + 31w + 78 The answer is 3w^2 + 31w + 78.

  7. A rectangular garden is drawn on a coordinate plane with corners at (2, 1), (8, 1), (8, 5), and (2, 5). If the gardener wants to plant flowers along the entire perimeter of the garden, how many units of fencing will she need to enclose the entire garden? Answer: 20 Solution: Identify the coordinates of the rectangle's corners. A = (2, 1) B = (8, 1) C = (8, 5) D = (2, 5) Understand the layout. Points A and B have the same y-coordinate (y = 1), so AB is a horizontal side.
    Full step-by-step solution

    Let's solve this step by step. Step 1: Identify the coordinates of the rectangle's corners. The points are: A = (2, 1) B = (8, 1) C = (8, 5) D = (2, 5) Step 2: Understand the layout. Points A and B have the same y-coordinate (y = 1), so AB is a horizontal side. Points B and C have the same x-coordinate (x = 8), so BC is a vertical side. Points C and D have the same y-coordinate (y = 5), so CD is a horizontal side. Points D and A have the same x-coordinate (x = 2), so DA is a vertical side. Step 3: Find the length of horizontal sides. From A (2, 1) to B (8, 1): Length = 8 - 2 = 6 units. From C (8, 5) to D (2, 5): Length = 8 - 2 = 6 units. Step 4: Find the length of vertical sides. From B (8, 1) to C (8, 5): Length = 5 - 1 = 4 units. From D (2, 5) to A (2, 1): Length = 5 - 1 = 4 units. Step 5: Calculate the perimeter. Perimeter = sum of all sides = 6 + 4 + 6 + 4. That is: 6 + 4 = 10, plus 6 = 16, plus 4 = 20. Step 6: Conclusion. The gardener needs 20 units of fencing to enclose the garden. Final answer: 20