Linear Expression Operations

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Linear Expression Operations · Grade 7 · Worksheet 2

1. Liam is designing a rectangular garden for his school. The length of the garden is represented by the expression (3x + 7) meters, and the width is (2x - 4) meters. To determine the amount of fencing he needs, he must calculate the perimeter. What is the simplified expression for the perimeter of Liam's garden? Answer: 10x + 6 Solution: To find the perimeter of a rectangle, we use the formula: Perimeter = 2 × (length + width) Write down the given expressions. Length = 3x + 7 Width = 2x - 4 Add the length and width.
   Full step-by-step solution

   To find the perimeter of a rectangle, we use the formula: Perimeter = 2 × (length + width) Step 1: Write down the given expressions. Length = 3x + 7 Width = 2x - 4 Step 2: Add the length and width. (3x + 7) + (2x - 4) = 3x + 2x + 7 - 4 = 5x + 3 Step 3: Multiply the sum by 2. 2 × (5x + 3) = 2 × 5x + 2 × 3 = 10x + 6 So the simplified expression for the perimeter is 10x + 6 meters.

2. A rectangular skatepark is drawn on a coordinate plane with vertices at (3, 1), (15, 1), (15, 9), and (3, 9). A triangular ramp section inside the park is marked with vertices at (3, 1), (15, 1), and (9, 5). Tane wants to find the area of the remaining flat skating area after removing the ramp section. What is that area? Answer: 72 Solution: Find the area of the rectangular skatepark. Length in x-direction: from x = 3 to x = 15 → 15 - 3 = 12 units. Height in y-direction: from y = 1 to y = 9 → 9 - 1 = 8 units.
   Full step-by-step solution

   Step 1: Find the area of the rectangular skatepark. Length in x-direction: from x = 3 to x = 15 → 15 - 3 = 12 units. Height in y-direction: from y = 1 to y = 9 → 9 - 1 = 8 units. Area of rectangle = length × height = 12 × 8 = 96 square units. Step 2: Find the area of the triangular ramp section. Vertices: A = (3, 1), B = (15, 1), C = (9, 5). Base of triangle: from (3, 1) to (15, 1) → 15 - 3 = 12 units. Height of triangle: from y = 1 (base) to y = 5 (vertex) → 5 - 1 = 4 units. Area of triangle = (1/2) × base × height = (1/2) × 12 × 4 = (1/2) × 48 = 24 square units. Step 3: Find the area of the remaining flat skating area. Remaining area = Total rectangle area - Triangle area = 96 - 24 = 72 square units. The answer is 72.

3. Maria is designing a community mural project that requires two different colored borders. The top border length is represented by (5x + 12) feet, while the bottom border is (3x - 8) feet. She also needs side borders: the left side is (2x + 5) feet and the right side is (4x - 3) feet. What is the simplified expression for the total length of all borders combined? Answer: 14x + 6 Solution: Write the expression for the total length by adding all border expressions: (5x + 12) + (3x - 8) + (2x + 5) + (4x - 3) Remove parentheses: 5x + 12 + 3x - 8 + 2x + 5 + 4x - 3 Combine like terms for x: 5x + 3x + 2x + 4x = 14x Combine constant terms: 12 - 8 + 5 - 3 = 4 + 5 - 3 = 9 - 3 = 6 Write the…
   Full step-by-step solution

   Step 1: Write the expression for the total length by adding all border expressions: (5x + 12) + (3x - 8) + (2x + 5) + (4x - 3) Step 2: Remove parentheses: 5x + 12 + 3x - 8 + 2x + 5 + 4x - 3 Step 3: Combine like terms for x: 5x + 3x + 2x + 4x = 14x Step 4: Combine constant terms: 12 - 8 + 5 - 3 = 4 + 5 - 3 = 9 - 3 = 6 Step 5: Write the simplified expression: 14x + 6 The total length of all borders is 14x + 6 feet.

4. Noah is helping his school's gardening club plant two rectangular flower beds. The length of the first flower bed is represented by the expression (8x + 19) feet, and its width is represented by (5x - 7) feet. The second flower bed has a length of (11x - 13) feet and a width of (9x + 14) feet. The club wants to install a decorative border around the perimeter of each flower bed. What simplified expression represents the total length of border needed for both flower beds combined? Answer: 66x + 26 Solution: Find the perimeter of the first flower bed. Perimeter of first bed = 2(length) + 2(width) = 2(8x + 19) + 2(5x - 7) = 16x + 38 + 10x - 14 = 16x + 10x + 38 - 14 = 26x + 24 feet Find the perimeter of the second flower bed.
   Full step-by-step solution

   Step 1: Find the perimeter of the first flower bed. Perimeter of first bed = 2(length) + 2(width) = 2(8x + 19) + 2(5x - 7) = 16x + 38 + 10x - 14 = 16x + 10x + 38 - 14 = 26x + 24 feet Step 2: Find the perimeter of the second flower bed. Perimeter of second bed = 2(length) + 2(width) = 2(11x - 13) + 2(9x + 14) = 22x - 26 + 18x + 28 = 22x + 18x - 26 + 28 = 40x + 2 feet Step 3: Add the two perimeters together to find the total border length. Total border = (26x + 24) + (40x + 2) = 26x + 40x + 24 + 2 = 66x + 26 feet The simplified expression for the total length of border needed is 66x + 26 feet.

5. Ava is tracking the total points her school's debate team has earned in three rounds. In the first round, they earned (8x + 22) points. In the second round, they earned (11x - 17) points. In the third round, they earned (6x + 14) points, but the judges later deducted a penalty of (3x - 9) points from this round's score. What simplified expression represents the total number of points the debate team has now? Answer: 22x + 28 Solution: Write the expression for the first round: (8x + 22) points. Write the expression for the second round: (11x - 17) points. The third round originally was (6x + 14) points, but a penalty of (3x - 9) points is deducted.
   Full step-by-step solution

   Step 1: Write the expression for the first round: (8x + 22) points. Step 2: Write the expression for the second round: (11x - 17) points. Step 3: The third round originally was (6x + 14) points, but a penalty of (3x - 9) points is deducted. Find the adjusted third-round score by subtracting the penalty: Adjusted third round = (6x + 14) - (3x - 9) = 6x + 14 - 3x + 9 = 6x - 3x + 14 + 9 = 3x + 23 points. Step 4: Add all three rounds' scores together to find the total: Total = (8x + 22) + (11x - 17) + (3x + 23) Step 5: Group like terms. First, group the terms with x: 8x, 11x, and 3x. Then, group the constant terms: 22, -17, and 23. Total = (8x + 11x + 3x) + (22 - 17 + 23) Step 6: Combine the x terms: 8x + 11x = 19x 19x + 3x = 22x Step 7: Combine the constant terms: 22 - 17 = 5 5 + 23 = 28 Step 8: Write the simplified expression. Total = 22x + 28 The simplified expression for the total number of points the debate team has now is 22x + 28.

6. Mason is designing a rectangular community garden on a coordinate grid. The garden has vertices at (12, 9), (29, 9), (29, 21), and (12, 21). Inside the garden, a rectangular vegetable patch is drawn with vertices at (15, 12), (26, 12), (26, 18), and (15, 18). A triangular flower bed is marked with vertices at (15, 12), (26, 12), and (20.5, 18). What is the area of the vegetable patch that is NOT covered by the flower bed? Answer: 33 Solution: Find the area of the rectangular vegetable patch. Length in x-direction: 26 - 15 = 11 units. Height in y-direction: 18 - 12 = 6 units.
   Full step-by-step solution

   Step 1: Find the area of the rectangular vegetable patch. Length in x-direction: 26 - 15 = 11 units. Height in y-direction: 18 - 12 = 6 units. Area = 11 × 6 = 66 square units. Step 2: Find the area of the triangular flower bed. The base is along the bottom from (15, 12) to (26, 12), so base = 26 - 15 = 11 units. The height is from y = 12 up to the top vertex at y = 18, so height = 18 - 12 = 6 units. Area of triangle = 1/2 × base × height = 1/2 × 11 × 6 = 33 square units. Step 3: Subtract the triangle area from the rectangle area to find the uncovered area: 66 - 33 = 33 square units. The answer is 33.