Linear Expression Operations

lessonbunny.com

Answer Key & Explanations

Linear Expression Operations · Grade 7 · Worksheet 3

1. A rectangular swimming pool is drawn on a coordinate plane with vertices at (15, 10), (45, 10), (45, 30), and (15, 30). A triangular shallow section for children is marked inside the pool with vertices at (15, 10), (45, 10), and (30, 20). What is the area of the deep swimming section of the pool? Answer: 450 Solution: Length in x-direction: from x = 15 to x = 45 → 45 - 15 = 30 units Height in y-direction: from y = 10 to y = 30 → 30 - 10 = 20 units Area of rectangle = length × height = 30 × 20 = 600 square units Vertices: A = (15, 10), B = (45, 10), C = (30, 20) Base of triangle: from (15, 10) to (45, 10) → 45…
   Full step-by-step solution

   Step 1: Find the area of the rectangular pool Length in x-direction: from x = 15 to x = 45 → 45 - 15 = 30 units Height in y-direction: from y = 10 to y = 30 → 30 - 10 = 20 units Area of rectangle = length × height = 30 × 20 = 600 square units Step 2: Find the area of the triangular shallow section Vertices: A = (15, 10), B = (45, 10), C = (30, 20) Base of triangle: from (15, 10) to (45, 10) → 45 - 15 = 30 units Height of triangle: from y = 10 to y = 20 → 20 - 10 = 10 units Area of triangle = (1/2) × base × height = (1/2) × 30 × 10 = 150 square units Step 3: Find the area of the deep swimming section Area of deep section = Total pool area - Shallow section area = 600 - 150 = 450 square units The answer is 450.

2. 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. The city council wants to add a protective frame around the entire mural. What simplified expression represents the total length of framing needed for the perimeter of Maria's mural? Answer: 16x + 10 Solution: The formula for perimeter of a rectangle is P = 2(length) + 2(width) Substitute the given expressions: P = 2(5x + 12) + 2(3x - 7) Distribute the 2: P = 10x + 24 + 6x - 14 Combine like terms: 10x + 6x = 16x and 24 - 14 = 10 The simplified expression is 16x + 10 Therefore, the total length of…
   Full step-by-step solution

   Step 1: The formula for perimeter of a rectangle is P = 2(length) + 2(width) Step 2: Substitute the given expressions: P = 2(5x + 12) + 2(3x - 7) Step 3: Distribute the 2: P = 10x + 24 + 6x - 14 Step 4: Combine like terms: 10x + 6x = 16x and 24 - 14 = 10 Step 5: The simplified expression is 16x + 10 Step 6: Therefore, the total length of framing needed is 16x + 10 feet

3. Noah is designing a rectangular mural on a coordinate grid, with vertices at (11, 16), (51, 16), (51, 36), and (11, 36). He plans to paint a triangular section with vertices at (11, 16), (51, 16), and (31, 26) in a different color. What is the area of the remaining rectangular section of the mural after the triangle is painted? Answer: 600 Solution: Find the area of the rectangular mural. Length (horizontal): 51 - 11 = 40 units Height (vertical): 36 - 16 = 20 units Area of rectangle = 40 * 20 = 800 square units Find the area of the triangular section.
   Full step-by-step solution

   Step 1: Find the area of the rectangular mural. Length (horizontal): 51 - 11 = 40 units Height (vertical): 36 - 16 = 20 units Area of rectangle = 40 * 20 = 800 square units Step 2: Find the area of the triangular section. Base of triangle: from (11, 16) to (51, 16) = 40 units Height of triangle: vertical distance from base (y=16) to the third vertex (y=26) = 26 - 16 = 10 units Area of triangle = (1/2) * base * height = (1/2) * 40 * 10 = 200 square units Step 3: Find the area of the remaining rectangular section. Remaining area = Rectangle area - Triangle area = 800 - 200 = 600 square units The answer is 600.

4. Charlotte is helping to organize the school's book donation drive. She is sorting books into two boxes. The number of books in the first box is represented by the expression (9x + 14) and the number of books in the second box is represented by (6x - 9). If Charlotte moves (4x + 5) books from the first box to the second box, what is the simplified expression for the total number of books now in the second box? Answer: 10x - 4 Solution: The second box originally has (6x - 9) books. Charlotte adds (4x + 5) books to the second box.
   Full step-by-step solution

   Step 1: The second box originally has (6x - 9) books. Step 2: Charlotte adds (4x + 5) books to the second box. Step 3: Write the sum: (6x - 9) + (4x + 5) Step 4: Remove parentheses: 6x - 9 + 4x + 5 Step 5: Combine like terms: (6x + 4x) + (-9 + 5) = 10x + (-4) = 10x - 4 The answer is 10x - 4.

5. A rectangular garden has a length of (3x + 5) meters and a width of (2x - 1) meters. A triangular flower bed is placed inside with a base of (x + 3) meters and a height of (x - 2) meters. What is the area of the remaining garden space after removing the flower bed? Answer: 5x² + 6x - 11 Solution: When working with geometric problems involving algebraic expressions, we use the same area formulas but substitute the expressions for measurements.
   Full step-by-step solution

   When working with geometric problems involving algebraic expressions, we use the same area formulas but substitute the expressions for measurements. The key is to carefully distribute terms when multiplying binomials and combine like terms when adding or subtracting expressions.

6. A rectangular city park is drawn on a coordinate plane with vertices at (20, 10), (80, 10), (80, 55), and (20, 55). A triangular playground is built inside the park with vertices at (20, 10), (80, 10), and (50, 40). What is the area of the park that is NOT taken up by the playground? Answer: 1800 Solution: Find the area of the rectangular park. Length (horizontal): from x = 20 to x = 80 = 80 - 20 = 60 units Height (vertical): from y = 10 to y = 55 = 55 - 10 = 45 units Area of rectangle = length * height = 60 * 45 = 2700 square units Find the area of the triangular playground.
   Full step-by-step solution

   Step 1: Find the area of the rectangular park. Length (horizontal): from x = 20 to x = 80 = 80 - 20 = 60 units Height (vertical): from y = 10 to y = 55 = 55 - 10 = 45 units Area of rectangle = length * height = 60 * 45 = 2700 square units Step 2: Find the area of the triangular playground. The triangle's base is along the bottom side from (20, 10) to (80, 10). Base length = 80 - 20 = 60 units Height of triangle: vertical distance from base at y = 10 to top vertex at y = 40 = 40 - 10 = 30 units Area of triangle = (1/2) * base * height = (1/2) * 60 * 30 = (1/2) * 1800 = 900 square units Step 3: Subtract the playground area from the park area. Remaining area = 2700 - 900 = 1800 square units The answer is 1800.

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.