Multi-Step Equations

lessonbunny.com

Answer Key & Explanations

Multi-Step Equations · Grade 7 · Worksheet 1

1. 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

2. 3(2x - 5) + 7 = 4x + 13 Answer: x = 10.5 Solution: 3(2x - 5) + 7 = 4x + 13 Distribute the 3 Multiply 3 by each term inside the parentheses: 3 * 2x = 6x 3 * (-5) = -15 6x - 15 + 7 = 4x + 13 -15 + 7 = -8 6x - 8 = 4x + 13 Subtract 4x from both sides: 6x - 4x - 8 = 4x - 4x + 13 2x - 8 = 13 Add 8 to both sides: 2x - 8 + 8 = 13 + 8 2x = 21 Divide both…
   Full step-by-step solution

   Let's solve the equation step-by-step. We start with: 3(2x - 5) + 7 = 4x + 13 --- **Step 1: Distribute the 3** Multiply 3 by each term inside the parentheses: 3 * 2x = 6x 3 * (-5) = -15 So the equation becomes: 6x - 15 + 7 = 4x + 13 --- **Step 2: Combine like terms on the left** -15 + 7 = -8 So now: 6x - 8 = 4x + 13 --- **Step 3: Move x terms to one side** Subtract 4x from both sides: 6x - 4x - 8 = 4x - 4x + 13 2x - 8 = 13 --- **Step 4: Move constant terms to the other side** Add 8 to both sides: 2x - 8 + 8 = 13 + 8 2x = 21 --- **Step 5: Solve for x** Divide both sides by 2: x = 21/2 x = 10.5 --- **Final answer:** x = 10.5

3. Sophia is organizing a community food drive. She has already collected 42 cans of food. She plans to collect more cans over the next several days. Each day, she will collect the same number of cans from local donors. After collecting for 8 days, she will donate 18 cans to a local shelter. If Sophia wants to have exactly 216 cans left to distribute to families in need, how many cans must she collect each day? Answer: 24 Solution: Let c represent the number of cans Sophia collects each day. Step 2: After 8 days, she will have collected 8c cans, plus the 42 she already has, for a total of 42 + 8c cans.
   Full step-by-step solution

   Step 1: Let c represent the number of cans Sophia collects each day. Step 2: After 8 days, she will have collected 8c cans, plus the 42 she already has, for a total of 42 + 8c cans. Step 3: She then donates 18 cans, so the number of cans left is (42 + 8c) - 18, which simplifies to 24 + 8c. Step 4: This amount must equal 216 cans. Write the equation: 24 + 8c = 216. Step 5: Subtract 24 from both sides: 8c = 216 - 24 = 192. Step 6: Divide both sides by 8: c = 192 / 8 = 24. Step 7: Sophia must collect 24 cans each day. The answer is 24.

4. 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.

5. 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

6. Emma is planning a school fundraiser and needs to order custom t-shirts. The printing company charges a $50 setup fee plus $8 per shirt. Emma's budget for the t-shirts is $650. If she wants to order the maximum number of shirts possible within her budget, how many t-shirts can she order? Answer: 75 Solution: Let x represent the number of t-shirts Emma can order. The total cost is the setup fee plus cost per shirt: 50 + 8x Set up the equation with her budget: 50 + 8x = 650 Subtract 50 from both sides: 8x = 600 Divide both sides by 8: x = 600 ÷ 8 Calculate: 600 ÷ 8 = 75 Emma can order 75 t-shirts…
   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 cost per shirt: 50 + 8x Step 3: Set up the equation with her budget: 50 + 8x = 650 Step 4: Subtract 50 from both sides: 8x = 600 Step 5: Divide both sides by 8: x = 600 ÷ 8 Step 6: Calculate: 600 ÷ 8 = 75 Emma can order 75 t-shirts within her budget.

7. Kaia is saving money to buy a new bicycle that costs $315. She already has $87 saved from her birthday. For the next 7 weeks, she plans to save the same amount each week from her allowance. After those 7 weeks, she will also use a $25 gift card from her grandmother. How much money does Kaia need to save each week to have exactly enough for the bicycle? Answer: 29 Solution: Let x be the amount Kaia saves each week. Total saved = $87 (birthday) + 7x (allowance) + $25 (gift card). Set up the equation: 87 + 7x + 25 = 315.
   Full step-by-step solution

   Step 1: Let x be the amount Kaia saves each week. Step 2: Total saved = $87 (birthday) + 7x (allowance) + $25 (gift card). Step 3: Set up the equation: 87 + 7x + 25 = 315. Step 4: Combine like terms: 112 + 7x = 315. Step 5: Subtract 112 from both sides: 7x = 203. Step 6: Divide both sides by 7: x = 29. Kaia needs to save $29 each week.