Multi-Step Equations

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Answer Key & Explanations

Multi-Step Equations · Grade 7 · Worksheet 2

1. Liam is saving money to buy a new video game that costs $65. He already has $28 saved from his allowance. For the next 4 weeks, he plans to save the same amount each week from his part-time job. How much money does Liam need to save each week to have exactly enough for the video game after 4 weeks? Answer: 9.25 Solution: Determine how much more money Liam needs. The game costs $65. He already has $28 saved.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Determine how much more money Liam needs.** The game costs $65. He already has $28 saved. Money still needed = 65 - 28 65 - 28 = 37 So, Liam needs $37 more. --- **Step 2: Understand the saving plan.** He will save the same amount each week for 4 weeks. Let the amount saved per week = \( x \). In 4 weeks, he will save \( 4 \times x \). --- **Step 3: Set up the equation.** The money saved in 4 weeks must equal $37. So: 4x = 37 --- **Step 4: Solve for x.** Divide both sides by 4: x = 37 / 4 37 ÷ 4 = 9.25 --- **Step 5: Interpret the result.** Liam needs to save $9.25 each week for 4 weeks to reach his goal. --- **Final answer:** 9.25

2. Mason is building a rectangular treehouse platform in his backyard. The length of the platform is 8 feet more than four times its width. If the perimeter of the platform is 116 feet, what is the width of the platform? Answer: 10 Solution: Let w represent the width of the platform in feet. The length is 4w + 8 feet. The formula for the perimeter of a rectangle is P = 2(length + width).
   Full step-by-step solution

   Step 1: Let w represent the width of the platform in feet. The length is 4w + 8 feet. Step 2: The formula for the perimeter of a rectangle is P = 2(length + width). Substitute the given perimeter and expressions: 116 = 2((4w + 8) + w). Step 3: Combine like terms inside the parentheses: (4w + 8) + w = 5w + 8. Step 4: The equation becomes 116 = 2(5w + 8). Step 5: Apply the distributive property: 116 = 10w + 16. Step 6: Subtract 16 from both sides: 116 - 16 = 10w, so 100 = 10w. Step 7: Divide both sides by 10: 100 / 10 = w, so w = 10. The width of the platform is 10 feet.

3. 3(x + 5) - 2(2x - 1) = 4 Answer: 13 Solution: 3(x + 5) - 2(2x - 1) = 4 3(x + 5) becomes 3*x + 3*5 = 3x + 15 -2(2x - 1) becomes -2*2x -2*(-1) = -4x + 2 3x + 15 - 4x + 2 = 4 3x - 4x = -1x (or -x) 15 + 2 = 17 -x + 17 = 4 Subtract 17 from both sides: -x + 17 - 17 = 4 - 17 -x = -13 Multiply both sides by -1: x = 13 Substitute x = 13 into the…
   Full step-by-step solution

   Let's solve the equation step by step. We start with: 3(x + 5) - 2(2x - 1) = 4 **Step 1: Distribute the numbers outside the parentheses** 3(x + 5) becomes 3*x + 3*5 = 3x + 15 -2(2x - 1) becomes -2*2x -2*(-1) = -4x + 2 So the equation becomes: 3x + 15 - 4x + 2 = 4 **Step 2: Combine like terms on the left side** 3x - 4x = -1x (or -x) 15 + 2 = 17 So now we have: -x + 17 = 4 **Step 3: Isolate the term with x** Subtract 17 from both sides: -x + 17 - 17 = 4 - 17 -x = -13 **Step 4: Solve for x** Multiply both sides by -1: x = 13 **Final check:** Substitute x = 13 into the original equation: 3(13 + 5) - 2(2*13 - 1) = 3(18) - 2(26 - 1) = 54 - 2(25) = 54 - 50 = 4 This matches the right-hand side, so the solution is correct. **Answer:** x = 13

4. 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.

5. Noah is designing a rectangular garden in his backyard. The length of the garden is 6 meters more than three times its width. If the perimeter of the garden is 76 meters, what is the width of the garden? Answer: 8 Solution: Let w represent the width of the garden in meters. The length is 6 meters more than three times the width, so length = 3w + 6. The perimeter of a rectangle is 2(length + width).
   Full step-by-step solution

   Step 1: Let w represent the width of the garden in meters. Step 2: The length is 6 meters more than three times the width, so length = 3w + 6. Step 3: The perimeter of a rectangle is 2(length + width). So the equation is 2(3w + 6 + w) = 76. Step 4: Combine like terms inside the parentheses: 3w + w = 4w. So 2(4w + 6) = 76. Step 5: Apply the distributive property: 2 * 4w = 8w and 2 * 6 = 12. So 8w + 12 = 76. Step 6: Subtract 12 from both sides: 8w + 12 - 12 = 76 - 12, so 8w = 64. Step 7: Divide both sides by 8: 8w / 8 = 64 / 8, so w = 8. The width of the garden is 8 meters.

6. 3(x + 5) - 7 = 2x + 14 Answer: x = 6 Solution: 3(x + 5) - 7 = 2x + 14 Distribute the 3 Multiply 3 by both terms inside the parentheses: 3 * x + 3 * 5 - 7 = 2x + 14 3x + 15 - 7 = 2x + 14 Combine 15 - 7: 3x + 8 = 2x + 14 Subtract 2x from both sides: 3x - 2x + 8 = 14 x + 8 = 14 Subtract 8 from both sides: x + 8 - 8 = 14 - 8 x = 6 Plug x = 6…
   Full step-by-step solution

   Let's solve the equation step-by-step. We start with: 3(x + 5) - 7 = 2x + 14 **Step 1: Distribute the 3** Multiply 3 by both terms inside the parentheses: 3 * x + 3 * 5 - 7 = 2x + 14 This gives: 3x + 15 - 7 = 2x + 14 **Step 2: Simplify the left side** Combine 15 - 7: 3x + 8 = 2x + 14 **Step 3: Move x terms to one side** Subtract 2x from both sides: 3x - 2x + 8 = 14 x + 8 = 14 **Step 4: Isolate x** Subtract 8 from both sides: x + 8 - 8 = 14 - 8 x = 6 **Step 5: Check the solution** Plug x = 6 into the original equation: 3(6 + 5) - 7 = 2(6) + 14 3(11) - 7 = 12 + 14 33 - 7 = 26 26 = 26 ✓ **Final answer:** x = 6

7. Mason is designing a rectangular flag on a coordinate grid. The flag has corners at (0, 0), (42, 0), (42, 27), and (0, 27). He draws a diagonal stripe from (0, 27) to (42, 0). The stripe is a triangle with vertices at (0, 27), (42, 0), and (0, 0). What is the area of the flag that is NOT covered by the triangular stripe? Answer: 567 Solution: Find the area of the rectangular flag. Length = 42 units (from x=0 to x=42). Width = 27 units (from y=0 to y=27).
   Full step-by-step solution

   Step 1: Find the area of the rectangular flag. Length = 42 units (from x=0 to x=42). Width = 27 units (from y=0 to y=27). Area = length x width = 42 x 27 = 1134 square units. Step 2: Find the area of the triangular stripe. The triangle has vertices at (0, 27), (42, 0), and (0, 0). Base of triangle = distance from (0,0) to (42,0) = 42 units. Height of triangle = vertical distance from base (y=0) to the top vertex at (0,27) = 27 units. Area of triangle = 1/2 x base x height = 1/2 x 42 x 27 = 21 x 27 = 567 square units. Step 3: Find the area of the flag not covered by the stripe. Remaining area = rectangle area - triangle area = 1134 - 567 = 567 square units. The answer is 567.

8. 5(3x - 8) + 12 = 3(2x + 7) - 4 Answer: 5 Solution: 5(3x - 8) + 12 = 3(2x + 7) - 4 15x - 40 + 12 = 6x + 21 - 4 15x - 28 = 6x + 17 Subtract 6x from both sides 15x - 28 - 6x = 6x + 17 - 6x 9x - 28 = 17 Add 28 to both sides 9x - 28 + 28 = 17 + 28 9x = 45 Divide both sides by 9 9x ÷ 9 = 45 ÷ 9 x = 5 The answer is 5.
   Full step-by-step solution

   Step 1: Distribute on both sides 5(3x - 8) + 12 = 3(2x + 7) - 4 15x - 40 + 12 = 6x + 21 - 4 Step 2: Combine like terms on each side 15x - 28 = 6x + 17 Step 3: Subtract 6x from both sides 15x - 28 - 6x = 6x + 17 - 6x 9x - 28 = 17 Step 4: Add 28 to both sides 9x - 28 + 28 = 17 + 28 9x = 45 Step 5: Divide both sides by 9 9x ÷ 9 = 45 ÷ 9 x = 5 The answer is 5.