Equivalent Expressions

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Answer Key & Explanations

Equivalent Expressions · Grade 7 · Worksheet 1

1. Rewrite 5(4x + 15) - 2(3x - 10) in simplest form. Answer: 14x + 95 Solution: 5 × 4x = 20x 5 × 15 = 75 So 5(4x + 15) = 20x + 75 Apply the distributive property to 2(3x - 10). 2 × 3x = 6x 2 × (-10) = -20 So 2(3x - 10) = 6x - 20 Subtract the second expression from the first.
   Full step-by-step solution

   Step 1: Apply the distributive property to 5(4x + 15). 5 × 4x = 20x 5 × 15 = 75 So 5(4x + 15) = 20x + 75 Step 2: Apply the distributive property to 2(3x - 10). 2 × 3x = 6x 2 × (-10) = -20 So 2(3x - 10) = 6x - 20 Step 3: Subtract the second expression from the first. (20x + 75) - (6x - 20) = 20x + 75 - 6x + 20 Step 4: Combine like terms. 20x - 6x = 14x 75 + 20 = 95 Final answer: 14x + 95

2. A rectangular garden has a length that is 3 meters more than twice its width. If the perimeter of the garden is 42 meters, what is the width of the garden in meters? Answer: 6 Solution: Let the width of the garden be \( w \) meters. The length is 3 meters more than twice the width, so: length \( l = 2w + 3 \). \( P = 2 \times (\text{length} + \text{width}) \).
   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 perimeter formula** The perimeter \( P \) of a rectangle is: \( P = 2 \times (\text{length} + \text{width}) \). Given \( P = 42 \), we have: \( 2 \times (l + w) = 42 \). --- **Step 3: Substitute the expression for length** Substitute \( l = 2w + 3 \) into the perimeter equation: \( 2 \times ( (2w + 3) + w ) = 42 \). --- **Step 4: Simplify inside the parentheses** \( (2w + 3) + w = 3w + 3 \). So: \( 2 \times (3w + 3) = 42 \). --- **Step 5: Solve for \( w \)** Divide both sides by 2: \( 3w + 3 = 21 \). Subtract 3 from both sides: \( 3w = 18 \). Divide by 3: \( w = 6 \). --- **Step 6: Conclusion** The width of the garden is 6 meters.

3. Isabella draws a rectangular prism with a length of 14 units, a width of 9 units, and a height of 11 units. She then creates a net of the prism by flattening all its faces. The net consists of six rectangles arranged in a cross-like pattern. Write two equivalent expressions that represent the total surface area of the prism: one in expanded form (showing the sum of the areas of all six faces) and one in factored form. Then determine if these two expressions are equivalent. Answer: Expanded: 2(14×9) + 2(14×11) + 2(9×11) = 252 + 308 + 198 = 758; Factored: 2(14×9 + 14×11 + 9×11) = 2(126 + 154 + 99) = 2(379) = 758. They are equivalent. Solution: Identify the dimensions of the rectangular prism: length (l) = 14 units, width (w) = 9 units, height (h) = 11 units.
   Full step-by-step solution

   Step 1: Identify the dimensions of the rectangular prism: length (l) = 14 units, width (w) = 9 units, height (h) = 11 units. Step 2: The prism has 6 faces: front and back (both l × h), left and right (both w × h), top and bottom (both l × w). Step 3: Expanded form: sum of areas of all six faces = 2(l × w) + 2(l × h) + 2(w × h) = 2(14 × 9) + 2(14 × 11) + 2(9 × 11). Step 4: Calculate each product: 14 × 9 = 126, 14 × 11 = 154, 9 × 11 = 99. Step 5: Expanded form: 2(126) + 2(154) + 2(99) = 252 + 308 + 198 = 758 square units. Step 6: Factored form: The common factor 2 can be factored out: 2(l × w + l × h + w × h) = 2(14 × 9 + 14 × 11 + 9 × 11). Step 7: Inside the parentheses: 126 + 154 + 99 = 379. Step 8: Factored form: 2(379) = 758 square units. Step 9: Both expressions evaluate to 758, so they are equivalent. The answer is: Expanded: 2(14×9) + 2(14×11) + 2(9×11) = 252 + 308 + 198 = 758; Factored: 2(14×9 + 14×11 + 9×11) = 2(126 + 154 + 99) = 2(379) = 758. They are equivalent.

4. Rewrite 4(6x + 8) - 12x in simplest form. Answer: 12x + 32 Solution: Distribute the 4 to both terms inside the parentheses: 4(6x + 8) = 4 × 6x + 4 × 8 = 24x + 32. Now the expression is 24x + 32 - 12x. Combine like terms: 24x - 12x = 12x.
   Full step-by-step solution

   Step 1: Distribute the 4 to both terms inside the parentheses: 4(6x + 8) = 4 × 6x + 4 × 8 = 24x + 32. Step 2: Now the expression is 24x + 32 - 12x. Step 3: Combine like terms: 24x - 12x = 12x. Step 4: The constant term 32 remains. Step 5: The simplified expression is 12x + 32. Final answer: 12x + 32

5. Emma is planning a school fundraiser and needs to order custom t-shirts. The printing company charges a flat setup fee of $125 plus $8.50 per shirt. If Emma's budget for t-shirts is $850, how many shirts can she order without exceeding her budget? Answer: 85 Solution: Let x represent the number of shirts Emma can order. The total cost is the setup fee plus cost per shirt: 125 + 8.50x Set up the inequality: 125 + 8.50x ≤ 850 Subtract 125 from both sides: 8.50x ≤ 725 Divide both sides by 8.50: x ≤ 725 ÷ 8.50 Calculate: 725 ÷ 8.50 = 85.29 Since Emma can't order…
   Full step-by-step solution

   Step 1: Let x represent the number of shirts Emma can order. Step 2: The total cost is the setup fee plus cost per shirt: 125 + 8.50x Step 3: Set up the inequality: 125 + 8.50x ≤ 850 Step 4: Subtract 125 from both sides: 8.50x ≤ 725 Step 5: Divide both sides by 8.50: x ≤ 725 ÷ 8.50 Step 6: Calculate: 725 ÷ 8.50 = 85.29 Step 7: Since Emma can't order a fraction of a shirt, she can order at most 85 shirts. The answer is 85 shirts.

6. Emma is planning a school fundraiser and needs to order custom t-shirts. The printing company charges a $75 setup fee plus $12 per shirt. If Emma's budget for t-shirts is $1,500, what is the maximum number of shirts she can order without exceeding her budget? Answer: 118 Solution: Let x represent the number of shirts Emma can order. The total cost is the setup fee plus the cost per shirt: 75 + 12x Set up the inequality: 75 + 12x ≤ 1500 Subtract 75 from both sides: 12x ≤ 1425 Divide both sides by 12: x ≤ 1425 ÷ 12 Calculate: 1425 ÷ 12 = 118.75 Since Emma cannot order a…
   Full step-by-step solution

   Step 1: Let x represent the number of shirts Emma can order. Step 2: The total cost is the setup fee plus the cost per shirt: 75 + 12x Step 3: Set up the inequality: 75 + 12x ≤ 1500 Step 4: Subtract 75 from both sides: 12x ≤ 1425 Step 5: Divide both sides by 12: x ≤ 1425 ÷ 12 Step 6: Calculate: 1425 ÷ 12 = 118.75 Step 7: Since Emma cannot order a fraction of a shirt, the maximum number is 118. The answer is 118 shirts.