Linear Expression Operations

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Linear Expression Operations · Grade 7 · Worksheet 2

1. Mason draws a rectangular garden on a coordinate plane with vertices at (17, 12), (37, 12), (37, 27), and (17, 27). Inside the garden, he marks a triangular flower bed with vertices at (17, 12), (37, 12), and (27, 22). He wants to plant grass in the remaining area of the garden. What is the area of the grassy region? Answer: 200 Solution: Find the area of the rectangular garden. Length (horizontal): 37 - 17 = 20 units Width (vertical): 27 - 12 = 15 units Area of rectangle = 20 x 15 = 300 square units Find the area of the triangular flower bed.
   Full step-by-step solution

   Step 1: Find the area of the rectangular garden. Length (horizontal): 37 - 17 = 20 units Width (vertical): 27 - 12 = 15 units Area of rectangle = 20 x 15 = 300 square units Step 2: Find the area of the triangular flower bed. The base of the triangle lies along the bottom edge from (17, 12) to (37, 12). Base = 37 - 17 = 20 units The height is the vertical distance from the base (y = 12) to the top vertex (y = 22). Height = 22 - 12 = 10 units Area of triangle = 1/2 x base x height = 1/2 x 20 x 10 = 1/2 x 200 = 100 square units Step 3: Subtract the area of the flower bed from the total garden area. Grassy area = 300 - 100 = 200 square units The answer is 200.

2. (24x - 18) - (10x + 14) = ? Answer: 14x - 32 Solution: Write the original expression: (24x - 18) - (10x + 14) Distribute the subtraction (negative sign) to each term in the second parentheses: 24x - 18 - 10x - 14 Combine the x terms: 24x - 10x = 14x Combine the constant terms: -18 - 14 = -32 Write the final simplified expression: 14x - 32 The answer…
   Full step-by-step solution

   Step 1: Write the original expression: (24x - 18) - (10x + 14) Step 2: Distribute the subtraction (negative sign) to each term in the second parentheses: 24x - 18 - 10x - 14 Step 3: Combine the x terms: 24x - 10x = 14x Step 4: Combine the constant terms: -18 - 14 = -32 Step 5: Write the final simplified expression: 14x - 32 The answer is 14x - 32.

3. Noah is tracking the total points his basketball team has scored this season. In the first game, they scored (9x + 17) points. In the second game, they scored (13x - 8) points. In the third game, they scored (2x + 25) points, but the coach realized that the score from the third game actually included a 5-point penalty that should be subtracted from their total. What is the simplified expression for the team's total points after removing the penalty from the third game? Answer: 24x + 29 Solution: Write the expression for the sum of all three games before removing the penalty. Total before penalty = (9x + 17) + (13x - 8) + (2x + 25) Remove the parentheses and group like terms.
   Full step-by-step solution

   Step 1: Write the expression for the sum of all three games before removing the penalty. Total before penalty = (9x + 17) + (13x - 8) + (2x + 25) Step 2: Remove the parentheses and group like terms. Total before penalty = 9x + 17 + 13x - 8 + 2x + 25 = (9x + 13x + 2x) + (17 - 8 + 25) Step 3: Combine the x-terms. 9x + 13x + 2x = 24x Step 4: Combine the constant terms. 17 - 8 + 25 = 9 + 25 = 34 So total before penalty = 24x + 34 Step 5: Now subtract the 5-point penalty. Final total = (24x + 34) - 5 = 24x + 34 - 5 = 24x + 29 The simplified expression for the team's total points after removing the penalty is 24x + 29.

4. Noah is tracking the total length of two hiking trails in a state park. The first trail has a length represented by (9x + 14) miles, and the second trail has a length represented by (7x - 11) miles. Park rangers decide to close a section of the second trail that measures (3x + 8) miles for maintenance. What simplified expression represents the total length of the hiking trails that remain open? Answer: 13x - 5 Solution: Write the expression for the first trail: (9x + 14) miles. The second trail originally is (7x - 11) miles, but a section of (3x + 8) miles is closed.
   Full step-by-step solution

   Step 1: Write the expression for the first trail: (9x + 14) miles. Step 2: The second trail originally is (7x - 11) miles, but a section of (3x + 8) miles is closed. Find the remaining length of the second trail by subtracting the closed section: Remaining second trail = (7x - 11) - (3x + 8) = 7x - 11 - 3x - 8 = 7x - 3x - 11 - 8 = 4x - 19 miles. Step 3: Add the first trail's length to the remaining second trail's length to find the total open length: Total = (9x + 14) + (4x - 19) = 9x + 14 + 4x - 19 = 9x + 4x + 14 - 19 = 13x - 5 miles. The simplified expression for the total length of trails that remain open is 13x - 5 miles.

5. Liam is designing a rectangular garden with a length of (3x + 5) meters and a width of (2x - 1) meters. He wants to add a decorative border along the entire perimeter. Write a simplified expression for the total length of border Liam needs in terms of x. Answer: 10x + 8 Solution: Length = (3x + 5) meters Width = (2x - 1) meters We need the total length of the border around the perimeter.
   Full step-by-step solution

   Step 1: Understand the problem We have a rectangular garden with: Length = (3x + 5) meters Width = (2x - 1) meters We need the total length of the border around the perimeter. Step 2: Recall the perimeter formula for a rectangle Perimeter P = 2 × (Length + Width) Step 3: Substitute the given expressions into the formula P = 2 × [ (3x + 5) + (2x - 1) ] Step 4: Simplify inside the brackets (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.

6. Liam is designing a rectangular garden with a length of (3x + 5) meters and a width of (2x - 1) meters. He wants to add a decorative border around the entire garden. Write a 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 into the formula.
   Full step-by-step solution

   Step 1: Recall the formula for the perimeter of a rectangle. The perimeter P of a rectangle is given by: P = 2 × (length + width) Step 2: Substitute the given expressions for length and width into the formula. 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 of Liam's garden is 10x + 8 meters.

7. Liam is designing a rectangular garden for his school project. The length of the garden is represented by the expression (4x + 7) meters, and the width is (2x - 3) meters. He needs to calculate the total length of fencing required to go around the entire garden. What expression represents the perimeter of Liam's garden in simplest form? Answer: 12x + 8 Solution: To find the perimeter of a rectangle, we use the formula: Perimeter = 2 × (length + width) Write down the given expressions for length and width. Length = 4x + 7 Width = 2x - 3 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 for length and width. Length = 4x + 7 Width = 2x - 3 Step 2: Add the length and width. (4x + 7) + (2x - 3) = 4x + 7 + 2x - 3 Step 3: Combine like terms. 4x + 2x = 6x 7 - 3 = 4 So, length + width = 6x + 4 Step 4: Multiply the sum by 2 to get the perimeter. Perimeter = 2 × (6x + 4) = 2 × 6x + 2 × 4 = 12x + 8 Final answer: The perimeter is 12x + 8 meters.