Expand Linear Expressions

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Expand Linear Expressions · Grade 7 · Worksheet 2

1. 2(3x + 4) - 5(x - 2) = ? Answer: x + 18 Solution: 2(3x + 4) - 5(x - 2) = ?
   Full step-by-step solution

   Let's solve step by step. We start with: 2(3x + 4) - 5(x - 2) = ? **Step 1: Apply the distributive property** First term: 2(3x + 4) = 2 * 3x + 2 * 4 = 6x + 8 Second term: -5(x - 2) = -5 * x + (-5) * (-2) = -5x + 10 So now we have: 6x + 8 - 5x + 10 **Step 2: Combine like terms** For the x terms: 6x - 5x = 1x (or just x) For the constant terms: 8 + 10 = 18 **Step 3: Write the final expression** x + 18 **Final answer:** x + 18

2. Mere is helping to organize the school's charity fun run. Each participant pays a registration fee of $15. Additionally, the school receives a fixed donation of $250 from a local sponsor, regardless of the number of participants. If p represents the number of participants, write a simplified expression using the distributive property to show the total amount of money the school receives from the fun run. Answer: 15p + 250 Solution: The total money is the registration fee per participant times the number of participants, plus the fixed donation. Write an expression: 15p + 250. This expression is already simplified, showing the total as 15p + 250.
   Full step-by-step solution

   Step 1: The total money is the registration fee per participant times the number of participants, plus the fixed donation. Step 2: Write an expression: 15p + 250. Step 3: This expression is already simplified, showing the total as 15p + 250. The answer is 15p + 250.

3. A rectangular community garden is being expanded. The original garden has a length of (4x + 7) meters and a width of (3x - 2) meters. The city wants to add a walking path around the entire garden that increases both dimensions by 5 meters. Write a simplified expression for the total area of the expanded garden including the walking path. Answer: 12x^2 + 71x + 24 Solution: Original length = (4x + 7) meters, original width = (3x - 2) meters The walking path adds 5 meters to both dimensions, so: New length = (4x + 7) + 5 = (4x + 12) meters New width = (3x - 2) + 5 = (3x + 3) meters Area of expanded garden = new length × new width Area = (4x + 12)(3x + 3) First: 4x ×…
   Full step-by-step solution

   Step 1: Original length = (4x + 7) meters, original width = (3x - 2) meters Step 2: The walking path adds 5 meters to both dimensions, so: New length = (4x + 7) + 5 = (4x + 12) meters New width = (3x - 2) + 5 = (3x + 3) meters Step 3: Area of expanded garden = new length × new width Area = (4x + 12)(3x + 3) Step 4: Use distributive property (FOIL method): First: 4x × 3x = 12x^2 Outer: 4x × 3 = 12x Inner: 12 × 3x = 36x Last: 12 × 3 = 36 Step 5: Combine like terms: 12x^2 + (12x + 36x) + 36 = 12x^2 + 48x + 36 Step 6: The simplified expression is 12x^2 + 48x + 36 square meters

4. Matiu is helping to organize a school fundraiser by setting up rectangular booths. Each booth has a length that is 12 meters more than twice its width. To accommodate more students, he decides to expand each booth by adding a 3-meter wide extension to both the length and the width. If the original width of a booth is represented by w meters, write a simplified expression in expanded form for the total area of the expanded booth (including the extension). Answer: 2w^2 + 22w + 60 Solution: Original width = w meters. Original length = 2w + 12 meters (12 more than twice the width). After adding the 3-meter extension to both dimensions: New width = w + 3 + 3 = w + 6 meters.
   Full step-by-step solution

   Step 1: Original width = w meters. Step 2: Original length = 2w + 12 meters (12 more than twice the width). Step 3: After adding the 3-meter extension to both dimensions: New width = w + 3 + 3 = w + 6 meters. New length = (2w + 12) + 3 + 3 = 2w + 18 meters. Step 4: Total area = (new length) × (new width) = (2w + 18)(w + 6). Step 5: Expand using the distributive property: (2w)(w) + (2w)(6) + (18)(w) + (18)(6) = 2w^2 + 12w + 18w + 108. Step 6: Combine like terms: 2w^2 + (12w + 18w) + 108 = 2w^2 + 30w + 108. The answer is 2w^2 + 30w + 108.

5. A rectangular swimming pool is drawn on a coordinate plane with corners at (3, 2), (15, 2), (15, 8), and (3, 8). A straight diagonal safety rope is installed from the bottom-left corner to the top-right corner. What is the length of this diagonal rope? Round your answer to the nearest tenth. Answer: 13.4 Solution: Identify the coordinates of the bottom-left and top-right corners: (3, 2) and (15, 8) Calculate the horizontal distance (width): 15 - 3 = 12 units Calculate the vertical distance (height): 8 - 2 = 6 units Apply the Pythagorean theorem: diagonal^2 = width^2 + height^2 diagonal^2 = 12^2 + 6^2 =…
   Full step-by-step solution

   Step 1: Identify the coordinates of the bottom-left and top-right corners: (3, 2) and (15, 8) Step 2: Calculate the horizontal distance (width): 15 - 3 = 12 units Step 3: Calculate the vertical distance (height): 8 - 2 = 6 units Step 4: Apply the Pythagorean theorem: diagonal^2 = width^2 + height^2 Step 5: diagonal^2 = 12^2 + 6^2 = 144 + 36 = 180 Step 6: diagonal = sqrt(180) = sqrt(36 × 5) = 6 × sqrt(5) Step 7: sqrt(5) ≈ 2.236, so diagonal ≈ 6 × 2.236 = 13.416 Step 8: Round to the nearest tenth: 13.4 Therefore, the length of the diagonal rope is 13.4 units.

6. 3(2x - 5) + 4(x + 3) = ? Answer: 10x - 3 Solution: 3(2x - 5) + 4(x + 3) Distribute the 3 into the first parentheses 3 * 2x = 6x 3 * (-5) = -15 So, 3(2x - 5) becomes 6x - 15. Distribute the 4 into the second parentheses 4 * x = 4x 4 * 3 = 12 So, 4(x + 3) becomes 4x + 12.
   Full step-by-step solution

   Let's solve the problem step by step. We start with: 3(2x - 5) + 4(x + 3) **Step 1: Distribute the 3 into the first parentheses** 3 * 2x = 6x 3 * (-5) = -15 So, 3(2x - 5) becomes 6x - 15. **Step 2: Distribute the 4 into the second parentheses** 4 * x = 4x 4 * 3 = 12 So, 4(x + 3) becomes 4x + 12. **Step 3: Write the expression after distribution** 6x - 15 + 4x + 12 **Step 4: Combine like terms** First, combine the x terms: 6x + 4x = 10x Then, combine the constant terms: -15 + 12 = -3 **Step 5: Write the final simplified expression** 10x - 3 **Final Answer:** 10x - 3

7. 3(4x - 7) + 2(5x + 3) = ? Answer: 22x - 15 Solution: Distribute 3 to both terms inside the first parentheses: 3 × 4x = 12x and 3 × (-7) = -21, so we get 12x - 21 Distribute 2 to both terms inside the second parentheses: 2 × 5x = 10x and 2 × 3 = 6, so we get 10x + 6 Combine all terms: (12x - 21) + (10x + 6) Combine like terms: 12x + 10x = 22x and…
   Full step-by-step solution

   Step 1: Distribute 3 to both terms inside the first parentheses: 3 × 4x = 12x and 3 × (-7) = -21, so we get 12x - 21 Step 2: Distribute 2 to both terms inside the second parentheses: 2 × 5x = 10x and 2 × 3 = 6, so we get 10x + 6 Step 3: Combine all terms: (12x - 21) + (10x + 6) Step 4: Combine like terms: 12x + 10x = 22x and -21 + 6 = -15 Step 5: The simplified expression is 22x - 15