Expand Linear Expressions

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Expand Linear Expressions · Grade 7 · Worksheet 1

1. Matiu is organizing a school fundraiser where he sells rectangular tiles decorated by students. Each tile has a length that is 4 centimeters more than twice its width. The width of each tile is represented by w centimeters. Matiu wants to create a special display by placing a uniform frame around each tile that adds 6 centimeters to both the length and the width. Write a simplified expression in expanded form for the total area of the framed tile, including the frame. Answer: 2w^2 + 28w + 96 Solution: Original width = w centimeters Original length = 2w + 4 centimeters (4 more than twice the width) The frame adds 6 centimeters to each side, so both length and width increase by 12 centimeters total (6 on each side).
   Full step-by-step solution

   Step 1: Original width = w centimeters Step 2: Original length = 2w + 4 centimeters (4 more than twice the width) Step 3: The frame adds 6 centimeters to each side, so both length and width increase by 12 centimeters total (6 on each side). Step 4: New width = w + 12 Step 5: New length = (2w + 4) + 12 = 2w + 16 Step 6: Area of framed tile = (new length) × (new width) = (2w + 16)(w + 12) Step 7: Expand using distributive property: (2w)(w) + (2w)(12) + (16)(w) + (16)(12) = 2w^2 + 24w + 16w + 192 Step 8: Combine like terms: 2w^2 + (24w + 16w) + 192 = 2w^2 + 40w + 192 The answer is 2w^2 + 40w + 192.

2. 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.

3. Emma is designing a rectangular mural for her school's art project. The mural's length is 5 meters more than three times its width. She needs to calculate the total area to determine how much paint to purchase. If the width of the mural is represented by w meters, write a simplified expression for the total area of the mural in expanded form. Answer: 3w^2 + 5w Solution: The width is w meters. The length is 5 meters more than three times the width, so length = 3w + 5. Area of a rectangle = length × width.
   Full step-by-step solution

   Step 1: The width is w meters. Step 2: The length is 5 meters more than three times the width, so length = 3w + 5. Step 3: Area of a rectangle = length × width. Step 4: Area = (3w + 5) × w Step 5: Apply the distributive property: (3w × w) + (5 × w) Step 6: Simplify: 3w^2 + 5w Step 7: The simplified expression for the total area is 3w^2 + 5w square meters.

4. A rectangular swimming pool is drawn on a coordinate plane with corners at (0, 0), (15, 0), (15, 8), and (0, 8). A straight diagonal safety rope is installed from the corner at (0, 0) to the opposite corner at (15, 8). What is the length of this diagonal rope? Round your answer to the nearest tenth of a unit. Answer: 17.0 Solution: Identify the coordinates of the diagonal endpoints: (0, 0) and (15, 8) Calculate the horizontal distance: 15 - 0 = 15 units Calculate the vertical distance: 8 - 0 = 8 units Apply the Pythagorean theorem: diagonal² = horizontal² + vertical² diagonal² = 15² + 8² = 225 + 64 = 289 diagonal = √289 =…
   Full step-by-step solution

   Step 1: Identify the coordinates of the diagonal endpoints: (0, 0) and (15, 8) Step 2: Calculate the horizontal distance: 15 - 0 = 15 units Step 3: Calculate the vertical distance: 8 - 0 = 8 units Step 4: Apply the Pythagorean theorem: diagonal² = horizontal² + vertical² Step 5: diagonal² = 15² + 8² = 225 + 64 = 289 Step 6: diagonal = √289 = 17 Step 7: Round to the nearest tenth: 17.0 Step 8: The length of the diagonal rope is 17.0 units.

5. Liam is designing a rectangular garden for his school's community project. The length of the garden is represented by the expression (3x + 5) meters, and the width is (2x - 1) meters. To calculate the amount of fencing needed, he needs to find the perimeter of the garden. What is the simplified expression for the perimeter of Liam's garden? Answer: 10x + 8 Solution: Recall the formula for the perimeter of a rectangle. P = 2 × (length + width) Substitute the given expressions for length and width.
   Full step-by-step solution

   Step 1: Recall the formula for the perimeter of a rectangle. The perimeter P is given by: P = 2 × (length + width) Step 2: Substitute the given expressions for length and width. Length = (3x + 5) meters Width = (2x - 1) meters So: P = 2 × [ (3x + 5) + (2x - 1) ] Step 3: Simplify inside the brackets first. Add the like terms: 3x + 2x = 5x 5 + (-1) = 4 So inside the brackets: 5x + 4 Step 4: Multiply by 2. P = 2 × (5x + 4) P = 2 × 5x + 2 × 4 P = 10x + 8 Step 5: Final simplified expression. The perimeter is 10x + 8 meters.

6. A rectangular garden is drawn on a coordinate plane with vertices at (2, 1), (8, 1), (8, 5), and (2, 5). If the gardener wants to expand the garden by applying a scale factor of 2.5 to all dimensions while keeping the bottom-left corner fixed at (2, 1), what will be the coordinates of the new top-right corner? Answer: (17, 11) Solution: (2, 1), (8, 1), (8, 5), (2, 5).
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understand the original rectangle** Vertices are given in order: (2, 1), (8, 1), (8, 5), (2, 5). Plotting these: - (2, 1) is bottom-left - (8, 1) is bottom-right - (8, 5) is top-right - (2, 5) is top-left So the original rectangle has: Width = 8 − 2 = 6 units Height = 5 − 1 = 4 units --- **Step 2: Identify the fixed point** The problem says: keep the bottom-left corner fixed at (2, 1). So (2, 1) is the anchor point for scaling. --- **Step 3: Apply the scale factor** Scale factor = 2.5 New width = 6 × 2.5 = 15 New height = 4 × 2.5 = 10 --- **Step 4: Find the new top-right corner** From the fixed bottom-left corner (2, 1): - New bottom-right corner = (2 + 15, 1) = (17, 1) - New top-left corner = (2, 1 + 10) = (2, 11) - New top-right corner = (2 + 15, 1 + 10) = (17, 11) --- **Step 5: Verify** Original top-right was (8, 5). If we scale distances from the fixed point (2, 1): Horizontal distance from fixed point: 8 − 2 = 6 → new horizontal distance = 6 × 2.5 = 15 → new x = 2 + 15 = 17 Vertical distance from fixed point: 5 − 1 = 4 → new vertical distance = 4 × 2.5 = 10 → new y = 1 + 10 = 11 So new top-right corner = (17, 11). --- **Final answer:** (17, 11)

7. A triangular park is drawn on a coordinate plane with vertices at (0, 0), (12, 0), and (0, 16). A straight walking path is drawn from the vertex at (0, 0) to the midpoint of the side connecting (12, 0) and (0, 16). What is the length of this walking path? Round your answer to the nearest tenth. Answer: 10.0 Solution: Find the midpoint between (12, 0) and (0, 16). Midpoint formula: ((x1 + x2)/2, (y1 + y2)/2) Midpoint = ((12 + 0)/2, (0 + 16)/2) = (12/2, 16/2) = (6, 8) Calculate the distance from (0, 0) to (6, 8).
   Full step-by-step solution

   Step 1: Find the midpoint between (12, 0) and (0, 16). Midpoint formula: ((x1 + x2)/2, (y1 + y2)/2) Midpoint = ((12 + 0)/2, (0 + 16)/2) = (12/2, 16/2) = (6, 8) Step 2: Calculate the distance from (0, 0) to (6, 8). Distance formula: sqrt((x2 - x1)^2 + (y2 - y1)^2) Distance = sqrt((6 - 0)^2 + (8 - 0)^2) = sqrt(6^2 + 8^2) = sqrt(36 + 64) = sqrt(100) = 10 Step 3: Round to the nearest tenth. 10.0 is already rounded to the nearest tenth. The length of the walking path is 10.0 units.