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Multi-Step Equations

Grade 7 · Algebra · Worksheet 3

  1. Emma is designing a rectangular mural for her school's art project. The length of the mural is 3 meters more than twice its width. If the perimeter of the mural is 54 meters, what is the width of the mural? Answer: ______________
  2. 3(2x - 7) + 5 = 4(x + 3) - 11 Answer: ______________
  3. Emma is planning a school fundraiser and needs to order custom t-shirts. The printing company charges a flat setup fee of $85 plus $12 per shirt. If Emma's budget for the t-shirts is $625, how many shirts can she order without exceeding her budget? Answer: ______________
  4. Aroha is organizing a school science fair. She needs to set up rectangular display boards in the gymnasium. The length of each display board is 9 meters more than three times its width. If the perimeter of the board is 50 meters, what is the width of the display board? Answer: ______________
  5. Matiu is saving money to buy a new laptop that costs $1,600. He already has $400 saved. He plans to save the same amount each week for the next 12 weeks from his part-time job. After those 12 weeks, his grandmother will give him $200 as a gift. How much does Matiu need to save each week to have exactly enough for the laptop? Answer: ______________
  6. Aroha is organizing a school talent show. She needs to set up chairs for the audience in rows. Each row has the same number of chairs. After setting up 9 rows, she has 15 chairs left over. If she had set up 11 rows instead, she would be 9 chairs short. How many chairs does Aroha have in total? Answer: ______________
  7. Olivia is organizing a school bake sale. She has already baked 35 cupcakes. She plans to bake more batches, and each batch will have 15 cupcakes. After baking all the batches, she will donate 25 cupcakes to a local shelter. If Olivia wants to have exactly 175 cupcakes left to sell at the bake sale, how many batches of cupcakes does she need to bake? Answer: ______________
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Answer Key & Explanations

Multi-Step Equations · Grade 7 · Worksheet 3

  1. Emma is designing a rectangular mural for her school's art project. The length of the mural is 3 meters more than twice its width. If the perimeter of the mural is 54 meters, what is the width of the mural? Answer: 8 Solution: Let w represent the width of the mural in meters. The length is 3 meters more than twice the width, so length = 2w + 3. The perimeter formula for a rectangle is P = 2(length + width).
    Full step-by-step solution

    Step 1: Let w represent the width of the mural in meters. Step 2: The length is 3 meters more than twice the width, so length = 2w + 3. Step 3: The perimeter formula for a rectangle is P = 2(length + width). Step 4: Substitute the known values: 54 = 2((2w + 3) + w) Step 5: Simplify inside the parentheses: 54 = 2(3w + 3) Step 6: Distribute the 2: 54 = 6w + 6 Step 7: Subtract 6 from both sides: 48 = 6w Step 8: Divide both sides by 6: w = 8 Step 9: The width of the mural is 8 meters.

  2. 3(2x - 7) + 5 = 4(x + 3) - 11 Answer: x = 5 Solution: Distribute on the left side: 3 × 2x = 6x and 3 × (-7) = -21, so left side becomes 6x - 21 + 5 Simplify left side: 6x - 21 + 5 = 6x - 16 Distribute on the right side: 4 × x = 4x and 4 × 3 = 12, so right side becomes 4x + 12 - 11 Simplify right side: 4x + 12 - 11 = 4x + 1 Equation is now: 6x - 16…
    Full step-by-step solution

    Step 1: Distribute on the left side: 3 × 2x = 6x and 3 × (-7) = -21, so left side becomes 6x - 21 + 5 Step 2: Simplify left side: 6x - 21 + 5 = 6x - 16 Step 3: Distribute on the right side: 4 × x = 4x and 4 × 3 = 12, so right side becomes 4x + 12 - 11 Step 4: Simplify right side: 4x + 12 - 11 = 4x + 1 Step 5: Equation is now: 6x - 16 = 4x + 1 Step 6: Subtract 4x from both sides: 6x - 4x - 16 = 1 → 2x - 16 = 1 Step 7: Add 16 to both sides: 2x - 16 + 16 = 1 + 16 → 2x = 17 Step 8: Divide both sides by 2: 2x ÷ 2 = 17 ÷ 2 → x = 8.5 The answer is x = 8.5.

  3. Emma is planning a school fundraiser and needs to order custom t-shirts. The printing company charges a flat setup fee of $85 plus $12 per shirt. If Emma's budget for the t-shirts is $625, how many shirts can she order without exceeding her budget? Answer: 45 Solution: Let x represent the number of shirts Emma can order. The total cost is the setup fee plus the cost per shirt: 85 + 12x Set up the equation with the budget: 85 + 12x ≤ 625 Subtract 85 from both sides: 12x ≤ 540 Divide both sides by 12: x ≤ 45 Since she can't order a fraction of a shirt, the…
    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: 85 + 12x Step 3: Set up the equation with the budget: 85 + 12x ≤ 625 Step 4: Subtract 85 from both sides: 12x ≤ 540 Step 5: Divide both sides by 12: x ≤ 45 Step 6: Since she can't order a fraction of a shirt, the maximum number is 45. Emma can order 45 shirts without exceeding her budget.

  4. Aroha is organizing a school science fair. She needs to set up rectangular display boards in the gymnasium. The length of each display board is 9 meters more than three times its width. If the perimeter of the board is 50 meters, what is the width of the display board? Answer: 4 Solution: Let the width be w meters. The length is 3w + 9 meters. The perimeter formula for a rectangle is P = 2l + 2w.
    Full step-by-step solution

    Step 1: Let the width be w meters. Step 2: The length is 3w + 9 meters. Step 3: The perimeter formula for a rectangle is P = 2l + 2w. Substitute the given perimeter and expressions: 50 = 2(3w + 9) + 2w. Step 4: Apply the distributive property: 50 = 6w + 18 + 2w. Step 5: Combine like terms: 50 = 8w + 18. Step 6: Subtract 18 from both sides: 50 - 18 = 8w + 18 - 18, so 32 = 8w. Step 7: Divide both sides by 8: 32/8 = 8w/8, so w = 4. The width of the display board is 4 meters.

  5. Matiu is saving money to buy a new laptop that costs $1,600. He already has $400 saved. He plans to save the same amount each week for the next 12 weeks from his part-time job. After those 12 weeks, his grandmother will give him $200 as a gift. How much does Matiu need to save each week to have exactly enough for the laptop? Answer: 83.33 Solution: Let x be the amount Matiu saves each week. Step 2: After 12 weeks, he saves 12x dollars. Step 3: Total money = initial savings + weekly savings + gift = 400 + 12x + 200.
    Full step-by-step solution

    Step 1: Let x be the amount Matiu saves each week. Step 2: After 12 weeks, he saves 12x dollars. Step 3: Total money = initial savings + weekly savings + gift = 400 + 12x + 200. Step 4: Set up the equation: 400 + 12x + 200 = 1600. Step 5: Combine like terms: 600 + 12x = 1600. Step 6: Subtract 600 from both sides: 12x = 1000. Step 7: Divide both sides by 12: x = 83.33... The answer is $83.33 (rounded to the nearest cent).

  6. Aroha is organizing a school talent show. She needs to set up chairs for the audience in rows. Each row has the same number of chairs. After setting up 9 rows, she has 15 chairs left over. If she had set up 11 rows instead, she would be 9 chairs short. How many chairs does Aroha have in total? Answer: 123 Solution: Let x be the number of chairs in each row. From the first scenario: Total chairs = 9x + 15 From the second scenario: Total chairs = 11x - 9 Set them equal: 9x + 15 = 11x - 9 Subtract 9x from both sides: 15 = 2x - 9 Add 9 to both sides: 24 = 2x Divide both sides by 2: x = 12 So each row has 12…
    Full step-by-step solution

    Let x be the number of chairs in each row. From the first scenario: Total chairs = 9x + 15 From the second scenario: Total chairs = 11x - 9 Set them equal: 9x + 15 = 11x - 9 Subtract 9x from both sides: 15 = 2x - 9 Add 9 to both sides: 24 = 2x Divide both sides by 2: x = 12 So each row has 12 chairs. Total chairs = 9(12) + 15 = 108 + 15 = 123. Aroha has 123 chairs.

  7. Olivia is organizing a school bake sale. She has already baked 35 cupcakes. She plans to bake more batches, and each batch will have 15 cupcakes. After baking all the batches, she will donate 25 cupcakes to a local shelter. If Olivia wants to have exactly 175 cupcakes left to sell at the bake sale, how many batches of cupcakes does she need to bake? Answer: 11 Solution: Let b represent the number of batches Olivia bakes. She bakes 15 cupcakes per batch, so she bakes 15b cupcakes. She starts with 35 cupcakes, so the total she has after baking is 35 + 15b.
    Full step-by-step solution

    Step 1: Let b represent the number of batches Olivia bakes. She bakes 15 cupcakes per batch, so she bakes 15b cupcakes. She starts with 35 cupcakes, so the total she has after baking is 35 + 15b. Step 2: She donates 25 cupcakes, so the number of cupcakes left to sell is (35 + 15b) - 25. This simplifies to 15b + 10. Step 3: We know this amount equals 175 cupcakes. So the equation is 15b + 10 = 175. Step 4: Subtract 10 from both sides: 15b + 10 - 10 = 175 - 10, which gives 15b = 165. Step 5: Divide both sides by 15: 15b / 15 = 165 / 15, which gives b = 11. Olivia needs to bake 11 batches of cupcakes.