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Linear Expression Operations

Grade 7 · Algebra · Worksheet 3

  1. Liam is designing a rectangular garden with a length of (3x + 5) meters and a width of (2x - 1) meters. He wants to install a decorative border around the entire garden. Write a simplified expression that represents the total length of border Liam needs to purchase. Answer: ______________
  2. Noah is designing a rectangular patio on a coordinate grid. The patio has vertices at (6, 11), (46, 11), (46, 31), and (6, 31). Inside the patio, he wants to build a triangular garden bed with vertices at (6, 11), (46, 11), and (26, 21). What is the area of the remaining patio space after the triangular garden bed is built? Answer: ______________
  3. Kaia is helping to organize the school's art supplies. She has a box of colored pencils that originally contains (9x + 14) pencils. She takes out (4x - 8) pencils to give to a group of students. Later, she finds another box with (6x + 10) pencils and adds them to the first box. What is the simplified expression for the total number of colored pencils Kaia has now? Answer: ______________
  4. Liam is designing a rectangular garden with a length of (3x + 7) meters and a width of (2x - 4) meters. He wants to add a decorative border along the entire perimeter. Write a simplified expression for the total length of border Liam needs to purchase. Answer: ______________
  5. Maria is designing a community mural project. The rectangular mural will have a length represented by (5x + 12) feet and a width represented by (3x - 7) feet. After planning, the community decides to extend the length by (2x + 5) feet to include more artwork, but reduce the width by (x - 3) feet to accommodate a walkway. What is the simplified expression for the perimeter of the final mural design? Answer: ______________
  6. Emma is designing a rectangular mural on a coordinate grid. The mural has vertices at (0, 0), (80, 0), (80, 50), and (0, 50). She wants to add a triangular accent section with vertices at (0, 0), (80, 0), and (40, 25). What is the area of the mural that is NOT part of the triangular accent? Answer: ______________
  7. (12x - 7) - (8x + 15) = ? Answer: ______________
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Answer Key & Explanations

Linear Expression Operations · Grade 7 · Worksheet 3

  1. Liam is designing a rectangular garden with a length of (3x + 5) meters and a width of (2x - 1) meters. He wants to install a decorative border around the entire garden. Write a simplified expression that represents the total length of border Liam needs to purchase. Answer: 10x + 8 Solution: Liam has a rectangular garden. The length is (3x + 5) meters and the width is (2x - 1) meters. He needs a border around the entire garden, which means we need the perimeter of the rectangle.
    Full step-by-step solution

    Step 1: Understand the problem Liam has a rectangular garden. The length is (3x + 5) meters and the width is (2x - 1) meters. He needs a border around the entire garden, which means we need the perimeter of the rectangle. Step 2: Recall the perimeter formula for a rectangle The formula for the perimeter P of a rectangle is: P = 2 * (length + width) Step 3: Substitute the given expressions into the formula Length = 3x + 5 Width = 2x - 1 So: P = 2 * ( (3x + 5) + (2x - 1) ) Step 4: Simplify inside the parentheses First, combine like terms inside: (3x + 5) + (2x - 1) = 3x + 2x + 5 - 1 = 5x + 4 Step 5: Multiply by 2 P = 2 * (5x + 4) = 2 * 5x + 2 * 4 = 10x + 8 Step 6: Final answer The total length of border needed is 10x + 8 meters.

  2. Noah is designing a rectangular patio on a coordinate grid. The patio has vertices at (6, 11), (46, 11), (46, 31), and (6, 31). Inside the patio, he wants to build a triangular garden bed with vertices at (6, 11), (46, 11), and (26, 21). What is the area of the remaining patio space after the triangular garden bed is built? Answer: 600 Solution: Find the area of the rectangular patio. Length in x-direction: from x = 6 to x = 46 gives 46 - 6 = 40 units. Height in y-direction: from y = 11 to y = 31 gives 31 - 11 = 20 units.
    Full step-by-step solution

    Step 1: Find the area of the rectangular patio. Length in x-direction: from x = 6 to x = 46 gives 46 - 6 = 40 units. Height in y-direction: from y = 11 to y = 31 gives 31 - 11 = 20 units. Area of rectangle = length × height = 40 × 20 = 800 square units. Step 2: Find the area of the triangular garden bed. The base is along the bottom from (6, 11) to (46, 11), so base = 46 - 6 = 40 units. The height is from y = 11 (base) up to the top vertex at y = 21, so height = 21 - 11 = 10 units. Area of triangle = (1/2) × base × height = (1/2) × 40 × 10 = (1/2) × 400 = 200 square units. Step 3: Subtract the triangle area from the rectangle area to find the remaining patio space: 800 - 200 = 600 square units. The answer is 600.

  3. Kaia is helping to organize the school's art supplies. She has a box of colored pencils that originally contains (9x + 14) pencils. She takes out (4x - 8) pencils to give to a group of students. Later, she finds another box with (6x + 10) pencils and adds them to the first box. What is the simplified expression for the total number of colored pencils Kaia has now? Answer: 11x + 32 Solution: Start with the original number of pencils: (9x + 14). Remove (4x - 8) pencils. This means subtract: (9x + 14) - (4x - 8).
    Full step-by-step solution

    Step 1: Start with the original number of pencils: (9x + 14). Step 2: Remove (4x - 8) pencils. This means subtract: (9x + 14) - (4x - 8). Step 3: Distribute the subtraction: 9x + 14 - 4x + 8 = (9x - 4x) + (14 + 8) = 5x + 22. Step 4: Add the new box of (6x + 10) pencils: (5x + 22) + (6x + 10). Step 5: Combine like terms: (5x + 6x) + (22 + 10) = 11x + 32. The simplified expression for the total number of pencils is 11x + 32.

  4. Liam is designing a rectangular garden with a length of (3x + 7) meters and a width of (2x - 4) meters. He wants to add a decorative border along the entire perimeter. Write a simplified expression for the total length of border Liam needs to purchase. Answer: 10x + 6 Solution: We have a rectangular garden. Length = (3x + 7) meters Width = (2x - 4) meters We need the total length of the border, which is the perimeter of the rectangle.
    Full step-by-step solution

    Step 1: Understand the problem We have a rectangular garden. Length = (3x + 7) meters Width = (2x - 4) meters We need the total length of the border, which is the perimeter of the rectangle. Step 2: Recall the formula for the perimeter of a rectangle Perimeter = 2 × (Length + Width) Step 3: Substitute the given expressions into the formula Perimeter = 2 × [ (3x + 7) + (2x - 4) ] Step 4: Simplify inside the brackets (3x + 7) + (2x - 4) = 3x + 2x + 7 - 4 = 5x + 3 Step 5: Multiply by 2 Perimeter = 2 × (5x + 3) = 2 × 5x + 2 × 3 = 10x + 6 Step 6: Final answer The total length of border Liam needs is 10x + 6 meters.

  5. Maria is designing a community mural project. The rectangular mural will have a length represented by (5x + 12) feet and a width represented by (3x - 7) feet. After planning, the community decides to extend the length by (2x + 5) feet to include more artwork, but reduce the width by (x - 3) feet to accommodate a walkway. What is the simplified expression for the perimeter of the final mural design? Answer: 18x + 20 Solution: Original length: (5x + 12) feet Extension: (2x + 5) feet New length = (5x + 12) + (2x + 5) = 5x + 12 + 2x + 5 = 7x + 17 feet Original width: (3x - 7) feet Reduction: (x - 3) feet New width = (3x - 7) - (x - 3) = 3x - 7 - x + 3 = 2x - 4 feet Perimeter = 2 × (length + width) Perimeter = 2 × [(7x +…
    Full step-by-step solution

    Step 1: Find the new length after extension Original length: (5x + 12) feet Extension: (2x + 5) feet New length = (5x + 12) + (2x + 5) = 5x + 12 + 2x + 5 = 7x + 17 feet Step 2: Find the new width after reduction Original width: (3x - 7) feet Reduction: (x - 3) feet New width = (3x - 7) - (x - 3) = 3x - 7 - x + 3 = 2x - 4 feet Step 3: Use the perimeter formula for a rectangle Perimeter = 2 × (length + width) Perimeter = 2 × [(7x + 17) + (2x - 4)] Step 4: Simplify inside the parentheses (7x + 17) + (2x - 4) = 7x + 2x + 17 - 4 = 9x + 13 Step 5: Multiply by 2 Perimeter = 2 × (9x + 13) = 18x + 26 The answer is 18x + 26.

  6. Emma is designing a rectangular mural on a coordinate grid. The mural has vertices at (0, 0), (80, 0), (80, 50), and (0, 50). She wants to add a triangular accent section with vertices at (0, 0), (80, 0), and (40, 25). What is the area of the mural that is NOT part of the triangular accent? Answer: 3000 Solution: Find the area of the rectangular mural. Length = 80 - 0 = 80 units. Height = 50 - 0 = 50 units.
    Full step-by-step solution

    Step 1: Find the area of the rectangular mural. Length = 80 - 0 = 80 units. Height = 50 - 0 = 50 units. Area of rectangle = 80 * 50 = 4000 square units. Step 2: Find the area of the triangular accent. The base of the triangle is from (0,0) to (80,0), so base = 80 units. The height of the triangle is the vertical distance from the base (y=0) to the third vertex (y=25), so height = 25 units. Area of triangle = (1/2) * base * height = (1/2) * 80 * 25 = (1/2) * 2000 = 1000 square units. Step 3: Subtract the triangle's area from the rectangle's area. Remaining area = 4000 - 1000 = 3000 square units. The answer is 3000.

  7. (12x - 7) - (8x + 15) = ? Answer: 4x - 22 Solution: Write the original expression: (12x - 7) - (8x + 15) Distribute the negative sign to both terms in the second parentheses: 12x - 7 - 8x - 15 Combine like terms for the x terms: 12x - 8x = 4x Combine like terms for the constants: -7 - 15 = -22 Write the final simplified expression: 4x - 22 The…
    Full step-by-step solution

    Step 1: Write the original expression: (12x - 7) - (8x + 15) Step 2: Distribute the negative sign to both terms in the second parentheses: 12x - 7 - 8x - 15 Step 3: Combine like terms for the x terms: 12x - 8x = 4x Step 4: Combine like terms for the constants: -7 - 15 = -22 Step 5: Write the final simplified expression: 4x - 22 The answer is 4x - 22.