Equivalent Expressions

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Equivalent Expressions · Grade 7 · Worksheet 2

1. Charlotte is designing a rectangular mural for her school's art show. The length of the mural is 14 meters more than 5 times its width. She needs to write two equivalent expressions for the area of the mural in expanded form and factored form to help calculate the amount of paint needed. If the width is w meters, write two equivalent expressions for the area. Then, find the area of the mural when the width is 11 meters using both expressions to verify they are equivalent. Answer: Area = 759 square meters Solution: Write the length in terms of the width. The length is 14 meters more than 5 times the width: L = 5w + 14. Write the area in expanded form.
   Full step-by-step solution

   Step 1: Write the length in terms of the width. The length is 14 meters more than 5 times the width: L = 5w + 14. Step 2: Write the area in expanded form. Area = length * width = (5w + 14) * w = 5w * w + 14 * w = 5w^2 + 14w. Step 3: Write the area in factored form. The common factor in 5w^2 + 14w is w: Area = w(5w + 14). Step 4: Substitute w = 11 into the expanded form: 5(11)^2 + 14(11) = 5(121) + 154 = 605 + 154 = 759. Step 5: Substitute w = 11 into the factored form: 11(5 * 11 + 14) = 11(55 + 14) = 11 * 69 = 759. Step 6: Both forms give 759, so they are equivalent. The area of the mural is 759 square meters.

2. Liam is designing a rectangular garden for his school's science project. The garden's length is 12.5 meters and its width is 8.4 meters. He needs to calculate the total area to determine how much soil to buy. What is the area of Liam's garden in square meters? Answer: 105 Solution: We are given a rectangular garden with length = 12.5 meters and width = 8.4 meters. The area of a rectangle is length × width. Area = 12.5 × 8.4 Think of 12.5 as 125/10 and 8.4 as 84/10.
   Full step-by-step solution

   Step 1: Understand the problem We are given a rectangular garden with length = 12.5 meters and width = 8.4 meters. The area of a rectangle is length × width. Step 2: Write the multiplication Area = 12.5 × 8.4 Step 3: Multiply ignoring decimals first Think of 12.5 as 125/10 and 8.4 as 84/10. So 125 × 84 = ? Step 4: Break down 125 × 84 125 × 80 = 125 × 8 × 10 = 1000 × 10 = 10,000 125 × 4 = 500 Add: 10,000 + 500 = 10,500 Step 5: Adjust for decimals 12.5 has 1 decimal place, 8.4 has 1 decimal place, so total decimal places = 2. Divide 10,500 by 100 (move decimal point two places left): 10,500 → 105.00 Step 6: Final answer Area = 105 square meters This means Liam needs enough soil to cover 105 square meters.

3. Rewrite 9(2x + 7) - 5x in simplest form. Answer: 13x + 63 Solution: Distribute the 9 to each term inside the parentheses: 9(2x + 7) = 9 × 2x + 9 × 7 = 18x + 63. Now the expression is 18x + 63 - 5x. Combine like terms (the x terms): 18x - 5x = 13x.
   Full step-by-step solution

   Step 1: Distribute the 9 to each term inside the parentheses: 9(2x + 7) = 9 × 2x + 9 × 7 = 18x + 63. Step 2: Now the expression is 18x + 63 - 5x. Step 3: Combine like terms (the x terms): 18x - 5x = 13x. Step 4: The simplified expression is 13x + 63. Final answer: 13x + 63.

4. Rewrite 3(5x + 7) - 2(3x - 5) in simplest form. Answer: 9x + 31 Solution: Expand the first group: 3(5x + 7) = 3*5x + 3*7 = 15x + 21 Expand the second group: -2(3x - 5) = -2*3x + (-2)*(-5) = -6x + 10 Combine the expanded forms: (15x + 21) + (-6x + 10) Combine like terms: 15x - 6x = 9x and 21 + 10 = 31 The simplest form is 9x + 31.
   Full step-by-step solution

   Step 1: Expand the first group: 3(5x + 7) = 3*5x + 3*7 = 15x + 21 Step 2: Expand the second group: -2(3x - 5) = -2*3x + (-2)*(-5) = -6x + 10 Step 3: Combine the expanded forms: (15x + 21) + (-6x + 10) Step 4: Combine like terms: 15x - 6x = 9x and 21 + 10 = 31 Step 5: The simplest form is 9x + 31. The answer is 9x + 31.

5. Rewrite 7(2x + 3) - 2(4x - 1) in simplest form. Answer: 6x + 23 Solution: Expand 7(2x + 3) using the distributive property: 7 × 2x = 14x and 7 × 3 = 21, so 7(2x + 3) = 14x + 21. Expand -2(4x - 1) using the distributive property: -2 × 4x = -8x and -2 × (-1) = +2, so -2(4x - 1) = -8x + 2.
   Full step-by-step solution

   Step 1: Expand 7(2x + 3) using the distributive property: 7 × 2x = 14x and 7 × 3 = 21, so 7(2x + 3) = 14x + 21. Step 2: Expand -2(4x - 1) using the distributive property: -2 × 4x = -8x and -2 × (-1) = +2, so -2(4x - 1) = -8x + 2. Step 3: Combine the two parts: (14x + 21) + (-8x + 2). Step 4: Combine like terms: 14x - 8x = 6x, and 21 + 2 = 23. Step 5: The simplified expression is 6x + 23. Final answer: 6x + 23.

6. Noah is planning a school fundraiser by selling tickets to a science fair. The cost to rent the venue is $850, and each ticket sells for $12. Noah needs to determine how many tickets must be sold to cover the venue cost and raise an additional $1,500 for new lab equipment. How many tickets does Noah need to sell? Answer: 196 Solution: Identify the total money needed from ticket sales. The venue cost is $850, and the fundraising goal is $1,500. Total money needed = $850 + $1,500 = $2,350.
   Full step-by-step solution

   Step 1: Identify the total money needed from ticket sales. The venue cost is $850, and the fundraising goal is $1,500. Total money needed = $850 + $1,500 = $2,350. Step 2: Each ticket contributes $12 towards this total. Number of tickets needed = Total money needed / Price per ticket. Number of tickets = $2,350 / $12. Step 3: Perform the division. $2,350 ÷ $12 = 195.833... Step 4: Since you cannot sell a fraction of a ticket, you must round up to the next whole number to ensure the goal is met. 195 tickets would raise 195 * $12 = $2,340, which is $10 short of $2,350. Therefore, 196 tickets are needed. Step 5: Verify. 196 tickets * $12 = $2,352. This covers the $850 venue cost ($2,352 - $850 = $1,502) and exceeds the $1,500 fundraising goal. The answer is 196.