Multi-Step Equations

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Multi-Step Equations · Grade 7 · Worksheet 1

1. A rectangular mural is painted on a wall with corners at (0, 0), (36, 0), (36, 20), and (0, 20). A triangular abstract design is painted inside the mural with vertices at (0, 0), (36, 0), and (18, 14). What is the area of the mural that is NOT covered by the triangular design? Answer: 468 Solution: Find the area of the rectangular mural. Length = 36 units (from x=0 to x=36) Width = 20 units (from y=0 to y=20) Area of rectangle = length × width = 36 × 20 = 720 square units Find the area of the triangular abstract design.
   Full step-by-step solution

   Step 1: Find the area of the rectangular mural. Length = 36 units (from x=0 to x=36) Width = 20 units (from y=0 to y=20) Area of rectangle = length × width = 36 × 20 = 720 square units Step 2: Find the area of the triangular abstract design. The triangle has vertices at (0, 0), (36, 0), and (18, 14). Base of triangle = distance from (0, 0) to (36, 0) = 36 units Height of triangle = vertical distance from base to point (18, 14) = 14 units Area of triangle = 1/2 × base × height = 1/2 × 36 × 14 = 18 × 14 = 252 square units Step 3: Find the area of the mural not covered by the triangle. Remaining area = rectangle area - triangle area = 720 - 252 = 468 square units The answer is 468.

2. Emma is planning a school fundraiser and needs to order custom t-shirts. The printing company charges a $150 setup fee plus $8 per shirt. If Emma's budget for the shirts is $1,230, how many t-shirts can she order? Answer: 135 Solution: Let x represent the number of t-shirts Emma can order. The total cost is the setup fee plus the cost per shirt: 150 + 8x Set up the equation with her budget: 150 + 8x = 1230 Subtract 150 from both sides: 8x = 1230 - 150 Calculate: 8x = 1080 Divide both sides by 8: x = 1080 ÷ 8 Calculate: x = 135…
   Full step-by-step solution

   Step 1: Let x represent the number of t-shirts Emma can order. Step 2: The total cost is the setup fee plus the cost per shirt: 150 + 8x Step 3: Set up the equation with her budget: 150 + 8x = 1230 Step 4: Subtract 150 from both sides: 8x = 1230 - 150 Step 5: Calculate: 8x = 1080 Step 6: Divide both sides by 8: x = 1080 ÷ 8 Step 7: Calculate: x = 135 Emma can order 135 t-shirts.

3. A factory produces 15,000 units of a product. After quality control inspection, 12% of the units are rejected. The remaining units are packaged into boxes that can hold 48 units each. How many full boxes can be packaged from the acceptable units? Answer: 275 Solution: Find the number of acceptable units after quality control. The factory produces 15,000 units. 12% are rejected, so 100% - 12% = 88% are acceptable.
   Full step-by-step solution

   Let's go step-by-step. **Step 1: Find the number of acceptable units after quality control.** The factory produces 15,000 units. 12% are rejected, so 100% - 12% = 88% are acceptable. Acceptable units = 88% of 15,000 = (88/100) × 15,000 = 0.88 × 15,000 = 13,200 **Step 2: Determine how many boxes are filled.** Each box holds 48 units. Number of full boxes = Total acceptable units ÷ Units per box = 13,200 ÷ 48 **Step 3: Perform the division.** 48 × 275 = 48 × (200 + 75) = 48 × 200 = 9,600; 48 × 75 = 3,600; total = 9,600 + 3,600 = 13,200 exactly. So 13,200 ÷ 48 = 275. **Step 4: Interpret the result.** Since 13,200 is exactly divisible by 48, all acceptable units fill 275 full boxes with no leftover units. **Final answer:** 275

4. 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.

5. Matiu is building a rectangular deck in his backyard. The length of the deck is 6 meters more than twice its width. The perimeter of the deck is 48 meters. What is the width of the deck? Answer: 6 Solution: Let w represent the width of the deck in meters. The length is 6 meters more than twice the width, so length = 2w + 6. The perimeter of a rectangle is P = 2(length + width).
   Full step-by-step solution

   Step 1: Let w represent the width of the deck in meters. Step 2: The length is 6 meters more than twice the width, so length = 2w + 6. Step 3: The perimeter of a rectangle is P = 2(length + width). Step 4: Substitute the given perimeter and expressions: 48 = 2((2w + 6) + w). Step 5: Simplify inside the parentheses: 2w + 6 + w = 3w + 6. Step 6: The equation becomes 48 = 2(3w + 6). Step 7: Apply the distributive property: 48 = 6w + 12. Step 8: Subtract 12 from both sides: 48 - 12 = 6w, so 36 = 6w. Step 9: Divide both sides by 6: w = 36 / 6 = 6. Step 10: The width of the deck is 6 meters. The answer is 6.

6. Mason is designing a rectangular playground on a coordinate grid. The playground has corners at (0, 0), (27, 0), (27, 17), and (0, 17). Inside the playground, a triangular sandbox is placed with vertices at (0, 0), (27, 0), and (13.5, 17). What is the area of the remaining grassy area outside the sandbox but inside the playground? Answer: 229.5 Solution: Find the area of the rectangular playground. Length = 27 - 0 = 27 units Width = 17 - 0 = 17 units Area of rectangle = length × width = 27 × 17 = 459 square units Find the area of the triangular sandbox.
   Full step-by-step solution

   Step 1: Find the area of the rectangular playground. Length = 27 - 0 = 27 units Width = 17 - 0 = 17 units Area of rectangle = length × width = 27 × 17 = 459 square units Step 2: Find the area of the triangular sandbox. The triangle has vertices at (0, 0), (27, 0), and (13.5, 17). Base of triangle = distance from (0, 0) to (27, 0) = 27 units Height of triangle = vertical distance from base at y = 0 to top vertex at y = 17 = 17 units Area of triangle = 1/2 × base × height = 1/2 × 27 × 17 = 229.5 square units Step 3: Find the remaining grassy area. Grassy area = rectangle area - triangle area = 459 - 229.5 = 229.5 square units The answer is 229.5.

7. Liam is saving money to buy a new video game that costs $65. He already has $27 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 game? Answer: 9.5 Solution: Figure out how much more money Liam needs. The game costs $65. He already has $27 saved.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Figure out how much more money Liam needs.** The game costs $65. He already has $27 saved. Money still needed = 65 - 27 65 - 27 = 38 So, Liam needs $38 more. --- **Step 2: Understand the saving plan.** He will save the same amount each week for 4 weeks. Let’s call the amount he saves each week \( x \). In 4 weeks, he will save \( 4 \times x \). This must equal the $38 he still needs. So: 4x = 38 --- **Step 3: Solve for x.** Divide both sides by 4: x = 38 / 4 38 ÷ 4 = 9.5 --- **Step 4: Interpret the result.** Liam needs to save $9.50 each week for 4 weeks. Check: Week 1: 27 + 9.5 = 36.5 Week 2: 36.5 + 9.5 = 46 Week 3: 46 + 9.5 = 55.5 Week 4: 55.5 + 9.5 = 65 ✅ --- **Final answer:** 9.5

8. 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