Multi-Step Equations

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Multi-Step Equations · Grade 7 · Worksheet 2

1. A school is organizing a field trip and needs to rent buses. Each bus can hold 42 students. There are 315 students going on the trip. The bus company charges a flat fee of $150 plus $85 per bus rented. How many buses does the school need to rent, and what will be the total cost? Answer: 8 buses, $830 Solution: Calculate how many buses are needed by dividing total students by bus capacity: 315 ÷ 42 = 7.5 Since you can't have half a bus, round up to the next whole number: 8 buses Calculate the cost for the buses: 8 buses × $85 per bus = $680 Add the flat fee: $680 + $150 = $830 The school needs 8 buses…
   Full step-by-step solution

   Step 1: Calculate how many buses are needed by dividing total students by bus capacity: 315 ÷ 42 = 7.5 Step 2: Since you can't have half a bus, round up to the next whole number: 8 buses Step 3: Calculate the cost for the buses: 8 buses × $85 per bus = $680 Step 4: Add the flat fee: $680 + $150 = $830 Step 5: The school needs 8 buses and the total cost will be $830.

2. A rectangular playground is drawn on a coordinate plane with corners at (2, 3), (2, 15), (10, 15), and (10, 3). A triangular sandbox is placed inside the playground with vertices at (2, 3), (10, 3), and (6, 11). The playground area that is not the sandbox needs to be covered with rubber mats. What is the area that needs mats? Answer: 64 Solution: Find the dimensions of the rectangular playground. Length = distance from x=2 to x=10 = 10 - 2 = 8 units Width = distance from y=3 to y=15 = 15 - 3 = 12 units Area of rectangle = length x width = 8 x 12 = 96 square units Find the area of the triangular sandbox.
   Full step-by-step solution

   Step 1: Find the dimensions of the rectangular playground. Length = distance from x=2 to x=10 = 10 - 2 = 8 units Width = distance from y=3 to y=15 = 15 - 3 = 12 units Area of rectangle = length x width = 8 x 12 = 96 square units Step 2: Find the area of the triangular sandbox. The base of the triangle is from (2,3) to (10,3). Base = 10 - 2 = 8 units. The height of the triangle is the vertical distance from the base (y=3) to the top vertex (y=11). Height = 11 - 3 = 8 units. Area of triangle = 1/2 x base x height = 1/2 x 8 x 8 = 32 square units Step 3: Find the area needing rubber mats. Mats area = rectangle area - triangle area = 96 - 32 = 64 square units. The answer is 64.

3. Emma is organizing a school bake sale. She bakes 3 batches of cookies, each batch containing the same number of cookies. After selling 17 cookies, she has 43 cookies left. How many cookies were in each batch? Answer: 20 Solution: Let x be the number of cookies in each batch. Step 2: Emma bakes 3 batches, so total cookies = 3x. Step 3: After selling 17, she has 43 left.
   Full step-by-step solution

   Step 1: Let x be the number of cookies in each batch. Step 2: Emma bakes 3 batches, so total cookies = 3x. Step 3: After selling 17, she has 43 left. Equation: 3x - 17 = 43. Step 4: Add 17 to both sides: 3x - 17 + 17 = 43 + 17, so 3x = 60. Step 5: Divide both sides by 3: 3x/3 = 60/3, so x = 20. The answer is 20 cookies per batch.

4. A rectangular garden is drawn on a coordinate plane with corners at (2, 1), (15, 1), (15, 8), and (2, 8). A triangular flower bed is placed inside the garden with vertices at (2, 1), (15, 1), and (8, 8). What is the area of the remaining garden space after removing the flower bed? Answer: 45.5 Solution: (2, 1), (15, 1), (15, 8), (2, 8). - The bottom side is from (2, 1) to (15, 1) → length = 15 - 2 = 13. - The top side is from (2, 8) to (15, 8) → length = 13.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understand the shapes** The rectangular garden has corners at: (2, 1), (15, 1), (15, 8), (2, 8). Plotting these: - The bottom side is from (2, 1) to (15, 1) → length = 15 - 2 = 13. - The top side is from (2, 8) to (15, 8) → length = 13. - The height is from y = 1 to y = 8 → height = 8 - 1 = 7. So the rectangle's area = length × height = 13 × 7 = 91. --- **Step 2: Identify the triangular flower bed** Vertices of the triangle: A = (2, 1), B = (15, 1), C = (8, 8). This triangle has base AB along the bottom of the rectangle from x = 2 to x = 15. Base length AB = 15 - 2 = 13. Height of triangle: The third vertex (8, 8) has y = 8, base is at y = 1, so height = 8 - 1 = 7. Area of triangle = (1/2) × base × height = (1/2) × 13 × 7. Compute: (1/2) × 91 = 45.5. --- **Step 3: Remaining garden area** Remaining area = area of rectangle − area of triangle = 91 − 45.5 = 45.5. --- **Final Answer:** 45.5

5. A rectangular swimming pool is drawn on a coordinate plane with corners at (0, 0), (20, 0), (20, 12), and (0, 12). A triangular diving area is marked off inside the pool with vertices at (0, 0), (20, 0), and (10, 8). What is the area of the remaining pool space that is not part of the diving area? Answer: 160 Solution: Length = 20 - 0 = 20 units Width = 12 - 0 = 12 units Area of rectangle = length × width = 20 × 12 = 240 square units The triangle has vertices at (0, 0), (20, 0), and (10, 8) Base of triangle = distance between (0, 0) and (20, 0) = 20 units Height of triangle = vertical distance from base to…
   Full step-by-step solution

   Step 1: Find the area of the rectangular pool Length = 20 - 0 = 20 units Width = 12 - 0 = 12 units Area of rectangle = length × width = 20 × 12 = 240 square units Step 2: Find the area of the triangular diving area The triangle has vertices at (0, 0), (20, 0), and (10, 8) Base of triangle = distance between (0, 0) and (20, 0) = 20 units Height of triangle = vertical distance from base to point (10, 8) = 8 units Area of triangle = 1/2 × base × height = 1/2 × 20 × 8 = 80 square units Step 3: Find the remaining pool area Remaining area = rectangle area - triangle area = 240 - 80 = 160 square units The answer is 160.

6. A rectangular swimming pool is drawn on a coordinate plane with corners at (0, 0), (20, 0), (20, 12), and (0, 12). A triangular diving area is marked off with vertices at (0, 0), (20, 0), and (10, 8). What is the area of the remaining pool surface that is NOT part of the diving area? Answer: 160 Solution: Length = 20 - 0 = 20 units Width = 12 - 0 = 12 units Area of rectangle = length × width = 20 × 12 = 240 square units The triangle has vertices at (0, 0), (20, 0), and (10, 8) Base of triangle = distance from (0, 0) to (20, 0) = 20 units Height of triangle = vertical distance from base to point…
   Full step-by-step solution

   Step 1: Find the area of the rectangular pool Length = 20 - 0 = 20 units Width = 12 - 0 = 12 units Area of rectangle = length × width = 20 × 12 = 240 square units Step 2: Find the area of the triangular diving area The triangle has vertices at (0, 0), (20, 0), and (10, 8) Base of triangle = distance from (0, 0) to (20, 0) = 20 units Height of triangle = vertical distance from base to point (10, 8) = 8 units Area of triangle = 1/2 × base × height = 1/2 × 20 × 8 = 80 square units Step 3: Find the remaining pool area Remaining area = rectangle area - triangle area = 240 - 80 = 160 square units The answer is 160.

7. 3(2x - 7) + 4 = 5(x + 2) - 8 Answer: x = 11 Solution: Distribute the 3 on the left side: 3 * 2x = 6x and 3 * (-7) = -21, so left side becomes 6x - 21 + 4 Distribute the 5 on the right side: 5 * x = 5x and 5 * 2 = 10, so right side becomes 5x + 10 - 8 Combine like terms on left side: -21 + 4 = -17, so left side is 6x - 17 Combine like terms on right…
   Full step-by-step solution

   Step 1: Distribute the 3 on the left side: 3 * 2x = 6x and 3 * (-7) = -21, so left side becomes 6x - 21 + 4 Step 2: Distribute the 5 on the right side: 5 * x = 5x and 5 * 2 = 10, so right side becomes 5x + 10 - 8 Step 3: Combine like terms on left side: -21 + 4 = -17, so left side is 6x - 17 Step 4: Combine like terms on right side: 10 - 8 = 2, so right side is 5x + 2 Step 5: Equation is now 6x - 17 = 5x + 2 Step 6: Subtract 5x from both sides: 6x - 5x - 17 = 2, which gives x - 17 = 2 Step 7: Add 17 to both sides: x - 17 + 17 = 2 + 17, which gives x = 19 Step 8: Check: 3(2*19 - 7) + 4 = 3(38 - 7) + 4 = 3(31) + 4 = 93 + 4 = 97, and 5(19 + 2) - 8 = 5(21) - 8 = 105 - 8 = 97 Both sides equal 97, so x = 19 is correct.

8. 3(x - 4) + 7 = 2x + 5 = ? Answer: 10 Solution: 3(x - 4) + 7 = 2x + 5 3(x - 4) means 3 times x and 3 times -4: 3*x - 3*4 + 7 = 2x + 5 3x - 12 + 7 = 2x + 5 -12 + 7 = -5 3x - 5 = 2x + 5 Subtract 2x from both sides: 3x - 2x - 5 = 2x - 2x + 5 x - 5 = 5 Add 5 to both sides: x - 5 + 5 = 5 + 5 x = 10 Substitute x = 10 into the original equation:…
   Full step-by-step solution

   Let's solve the equation step by step. We start with: 3(x - 4) + 7 = 2x + 5 **Step 1: Expand the left side** 3(x - 4) means 3 times x and 3 times -4: 3*x - 3*4 + 7 = 2x + 5 This simplifies to: 3x - 12 + 7 = 2x + 5 **Step 2: Combine like terms on the left** -12 + 7 = -5 So we have: 3x - 5 = 2x + 5 **Step 3: Get all x terms on one side** Subtract 2x from both sides: 3x - 2x - 5 = 2x - 2x + 5 x - 5 = 5 **Step 4: Isolate x** Add 5 to both sides: x - 5 + 5 = 5 + 5 x = 10 **Step 5: Check the solution** Substitute x = 10 into the original equation: 3(10 - 4) + 7 = 3(6) + 7 = 18 + 7 = 25 2(10) + 5 = 20 + 5 = 25 Both sides are equal, so the solution is correct. **Final answer:** x = 10