Expand Linear Expressions

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Answer Key & Explanations

Expand Linear Expressions · Grade 7 · Worksheet 3

1. A rectangular prism is drawn on a coordinate plane with vertices at (0, 0, 0), (6, 0, 0), (6, 4, 0), (0, 4, 0), (0, 0, 3), (6, 0, 3), (6, 4, 3), and (0, 4, 3). What is the total surface area of this rectangular prism? Answer: 108 Solution: Identify the dimensions from the coordinates. The x-coordinate goes from 0 to 6, so length = 6 units. The y-coordinate goes from 0 to 4, so width = 4 units.
   Full step-by-step solution

   Step 1: Identify the dimensions from the coordinates. The x-coordinate goes from 0 to 6, so length = 6 units. The y-coordinate goes from 0 to 4, so width = 4 units. The z-coordinate goes from 0 to 3, so height = 3 units. Step 2: A rectangular prism has 6 faces: 2 of each type (length×width, length×height, width×height). Step 3: Calculate area of length×width faces: 6 × 4 = 24. There are 2 of these: 2 × 24 = 48. Step 4: Calculate area of length×height faces: 6 × 3 = 18. There are 2 of these: 2 × 18 = 36. Step 5: Calculate area of width×height faces: 4 × 3 = 12. There are 2 of these: 2 × 12 = 24. Step 6: Add all face areas: 48 + 36 + 24 = 108. The total surface area is 108 square units.

2. 4(3x - 7) + 2(5x + 6) = ? Answer: 22x - 16 Solution: Apply distributive property to the first term: 4(3x - 7) = 4 × 3x + 4 × (-7) = 12x - 28 Apply distributive property to the second term: 2(5x + 6) = 2 × 5x + 2 × 6 = 10x + 12 Combine all terms: (12x - 28) + (10x + 12) = 12x + 10x - 28 + 12 Combine like terms: 12x + 10x = 22x and -28 + 12 = -16…
   Full step-by-step solution

   Step 1: Apply distributive property to the first term: 4(3x - 7) = 4 × 3x + 4 × (-7) = 12x - 28 Step 2: Apply distributive property to the second term: 2(5x + 6) = 2 × 5x + 2 × 6 = 10x + 12 Step 3: Combine all terms: (12x - 28) + (10x + 12) = 12x + 10x - 28 + 12 Step 4: Combine like terms: 12x + 10x = 22x and -28 + 12 = -16 Step 5: Final simplified expression: 22x - 16

3. Mere is organizing the school's book fair in the library. She has a rectangular display table. The length of the table is (8x + 12) centimeters and the width is (6x - 10) centimeters. To fit all the books, she needs to cover the entire table with a special cloth. Write a simplified expanded expression for the area of the table in square centimeters. Answer: 48x^2 + 32x - 120 Solution: Area of a rectangle = length × width Area = (8x + 12)(6x - 10) Use the distributive property: (8x)(6x) + (8x)(-10) + (12)(6x) + (12)(-10) Multiply each pair: 48x^2 + (-80x) + 72x + (-120) Combine like terms: 48x^2 + (-80x + 72x) - 120 = 48x^2 + (-8x) - 120 Simplified expression: 48x^2 - 8x - 120…
   Full step-by-step solution

   Step 1: Area of a rectangle = length × width Step 2: Area = (8x + 12)(6x - 10) Step 3: Use the distributive property: (8x)(6x) + (8x)(-10) + (12)(6x) + (12)(-10) Step 4: Multiply each pair: 48x^2 + (-80x) + 72x + (-120) Step 5: Combine like terms: 48x^2 + (-80x + 72x) - 120 = 48x^2 + (-8x) - 120 Step 6: Simplified expression: 48x^2 - 8x - 120 The answer is 48x^2 - 8x - 120.

4. 3(2x + 5) - 4(x - 3) = ? Answer: 2x + 27 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.
   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: Rewrite the expression with the distributed terms** 6x + 15 - 4x + 12 **Step 4: Combine like terms** For the x terms: 6x - 4x = 2x For the constant terms: 15 + 12 = 27 **Step 5: Write the final simplified expression** 2x + 27 **Final answer:** 2x + 27

5. Liam is designing a rectangular garden for his school's community project. The garden's length is 3 meters more than twice its width. If the total area of the garden needs to be 35 square meters, what is the width of the garden in meters? Answer: 3.5 Solution: Let the width of the garden be \( w \) meters. The length is 3 meters more than twice the width, so: length \( l = 2w + 3 \). Area of a rectangle = length × width.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Define variables** Let the width of the garden be \( w \) meters. The length is 3 meters more than twice the width, so: length \( l = 2w + 3 \). --- **Step 2: Write the area equation** Area of a rectangle = length × width. Given area = 35 square meters: \[ (2w + 3) \times w = 35 \] --- **Step 3: Expand and rearrange** \[ 2w^2 + 3w = 35 \] Subtract 35 from both sides: \[ 2w^2 + 3w - 35 = 0 \] --- **Step 4: Solve the quadratic equation** We can use the quadratic formula: \( w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 2 \), \( b = 3 \), \( c = -35 \). First, discriminant \( D \): \[ D = b^2 - 4ac = 3^2 - 4(2)(-35) = 9 + 280 = 289 \] \[ \sqrt{D} = \sqrt{289} = 17 \] --- **Step 5: Apply the formula** \[ w = \frac{-3 \pm 17}{2 \times 2} = \frac{-3 \pm 17}{4} \] Two possible solutions: \[ w = \frac{-3 + 17}{4} = \frac{14}{4} = 3.5 \] \[ w = \frac{-3 - 17}{4} = \frac{-20}{4} = -5 \] --- **Step 6: Interpret the results** Width cannot be negative, so we discard \( w = -5 \). Thus, the width is \( w = 3.5 \) meters. --- **Final answer:** 3.5

6. 4(2x - 7) + 3(5x + 4) = ? Answer: 23x - 16 Solution: Distribute 4 to both terms inside the first parentheses: 4 × 2x = 8x and 4 × (-7) = -28 Distribute 3 to both terms inside the second parentheses: 3 × 5x = 15x and 3 × 4 = 12 Write the expanded expression: 8x - 28 + 15x + 12 Combine like terms for x: 8x + 15x = 23x Combine constant terms: -28 +…
   Full step-by-step solution

   Step 1: Distribute 4 to both terms inside the first parentheses: 4 × 2x = 8x and 4 × (-7) = -28 Step 2: Distribute 3 to both terms inside the second parentheses: 3 × 5x = 15x and 3 × 4 = 12 Step 3: Write the expanded expression: 8x - 28 + 15x + 12 Step 4: Combine like terms for x: 8x + 15x = 23x Step 5: Combine constant terms: -28 + 12 = -16 Step 6: Write the final simplified expression: 23x - 16 The answer is 23x - 16.

7. A rectangular garden has a length of (3x + 5) meters and a width of (2x - 1) meters. The gardener wants to install a decorative border around the entire garden. Write a simplified expression that represents the total perimeter of the garden in meters. 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 parentheses first. (3x + 5) + (2x - 1) = 3x + 2x + 5 - 1 = 5x + 4 Step 4: Multiply the simplified expression by 2. P = 2 * (5x + 4) = 2 * 5x + 2 * 4 = 10x + 8 Step 5: Final simplified expression for the perimeter. The perimeter is 10x + 8 meters.

8. Aroha is organizing a school fundraiser by selling tickets to a cultural festival. Each student is required to sell tickets in packs. The number of tickets in each pack is represented by the expression (4t + 9), and each student sells (3t - 5) packs. What is the simplified expression, in expanded form, for the total number of tickets sold by one student? Answer: 12t^2 + 7t - 45 Solution: Write the expressions: number of tickets per pack = 4t + 9, number of packs = 3t - 5. Total tickets = (4t + 9)(3t - 5).
   Full step-by-step solution

   Step 1: Write the expressions: number of tickets per pack = 4t + 9, number of packs = 3t - 5. Step 2: Total tickets = (4t + 9)(3t - 5). Step 3: Use the distributive property (FOIL method): First: 4t * 3t = 12t^2 Outer: 4t * (-5) = -20t Inner: 9 * 3t = 27t Last: 9 * (-5) = -45 Step 4: Combine like terms: -20t + 27t = 7t Step 5: The expanded expression is 12t^2 + 7t - 45. The answer is 12t^2 + 7t - 45.