One Variable Equations

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Answer Key & Explanations

One Variable Equations · Grade 6 · Worksheet 3

1. Emma is planning a school fundraiser and needs to buy supplies. She has a budget of $350. For decorations, she spends $125. She then buys snacks that cost $2.50 per person for 60 people. How much money does Emma have left in her budget after these purchases? Answer: 75 Solution: Calculate the cost of snacks: $2.50 per person × 60 people = $150 Calculate total spending: decorations $125 + snacks $150 = $275 Subtract total spending from budget: $350 - $275 = $75 Emma has $75 left in her budget.
   Full step-by-step solution

   Step 1: Calculate the cost of snacks: $2.50 per person × 60 people = $150 Step 2: Calculate total spending: decorations $125 + snacks $150 = $275 Step 3: Subtract total spending from budget: $350 - $275 = $75 Emma has $75 left in her budget.

2. 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 dog-walking job. How much money does Liam need to save each week to have exactly enough for the video game? Answer: 9.25 Solution: The video game costs $65. Liam already has $28 saved. He will save the same amount each week for 4 weeks.
   Full step-by-step solution

   Let's go step by step. --- **Step 1: Understand the problem** The video game costs $65. Liam already has $28 saved. He will save the same amount each week for 4 weeks. We need to find how much he must save per week to have exactly $65. --- **Step 2: Find how much more money he needs** Money needed = Cost of game − Money already saved Money needed = 65 − 28 Money needed = 37 So Liam needs $37 more. --- **Step 3: Set up the weekly savings** Let \( x \) = amount saved per week. He saves for 4 weeks, so total from these weeks = \( 4 \times x \). We want: Money from weeks + Money already saved = Cost of game \( 4x + 28 = 65 \) --- **Step 4: Solve for \( x \)** \( 4x + 28 = 65 \) Subtract 28 from both sides: \( 4x = 65 - 28 \) \( 4x = 37 \) Divide both sides by 4: \( x = 37 / 4 \) --- **Step 5: Calculate the division** \( 37 / 4 = 9.25 \) --- **Step 6: Conclusion** Liam needs to save $9.25 each week for 4 weeks. --- **Final answer:** 9.25

3. 3(2x - 7) + 4 = 25 Answer: 7 Solution: Step 1: Distribute the 3: 3 × 2x = 6x and 3 × (-7) = -21 Step 2: Rewrite the equation: 6x - 21 + 4 = 25 Step 3: Combine like terms: 6x - 17 = 25 Step 4: Add 17 to both sides: 6x = 42 Step 5: Divide both sides by 6: x = 7 Step 6: Verify: 3(2×7 - 7) + 4 = 3(14 - 7) + 4 = 3×7 + 4 = 21 + 4 = 25 The…
   Full step-by-step solution

   Step 1: Distribute the 3: 3 × 2x = 6x and 3 × (-7) = -21 Step 2: Rewrite the equation: 6x - 21 + 4 = 25 Step 3: Combine like terms: 6x - 17 = 25 Step 4: Add 17 to both sides: 6x = 42 Step 5: Divide both sides by 6: x = 7 Step 6: Verify: 3(2×7 - 7) + 4 = 3(14 - 7) + 4 = 3×7 + 4 = 21 + 4 = 25 The answer is 7.

4. x + 1272 = 2807. Solve for x. Answer: 1535 Solution: The equation is x + 1272 = 2807. To isolate x, subtract 1272 from both sides of the equation: x + 1272 - 1272 = 2807 - 1272. Simplify: x = 1535.
   Full step-by-step solution

   Step 1: The equation is x + 1272 = 2807. Step 2: To isolate x, subtract 1272 from both sides of the equation: x + 1272 - 1272 = 2807 - 1272. Step 3: Simplify: x = 1535. Step 4: Check: 1535 + 1272 = 2807. The answer is 1535.

5. 3(x - 4) + 2x = 28 Answer: 8 Solution: Step 1: Distribute the 3: 3(x - 4) = 3x - 12 Step 2: Rewrite the equation: 3x - 12 + 2x = 28 Step 3: Combine like terms: 5x - 12 = 28 Step 4: Add 12 to both sides: 5x = 40 Step 5: Divide both sides by 5: x = 8 Step 6: Verify: 3(8 - 4) + 2(8) = 3(4) + 16 = 12 + 16 = 28 The answer is 8.
   Full step-by-step solution

   Step 1: Distribute the 3: 3(x - 4) = 3x - 12 Step 2: Rewrite the equation: 3x - 12 + 2x = 28 Step 3: Combine like terms: 5x - 12 = 28 Step 4: Add 12 to both sides: 5x = 40 Step 5: Divide both sides by 5: x = 8 Step 6: Verify: 3(8 - 4) + 2(8) = 3(4) + 16 = 12 + 16 = 28 The answer is 8.

6. x + 2849 = 5723 Answer: 2874 Solution: The equation is x + 2849 = 5723. To isolate x, subtract 2849 from both sides. x + 2849 - 2849 = 5723 - 2849 x = 2874 The answer is 2874.
   Full step-by-step solution

   Step 1: The equation is x + 2849 = 5723. To isolate x, subtract 2849 from both sides. Step 2: x + 2849 - 2849 = 5723 - 2849 Step 3: x = 2874 The answer is 2874.

7. Liam is designing a rectangular garden for his school project. The length of the garden is 12 meters, and the perimeter is 38 meters. He needs to find the width of the garden to calculate how much fencing to buy. What is the width of the garden? Answer: 7 Solution: Recall the perimeter formula for a rectangle. P = 2 × (length + width) Write down the given values. Length = 12 meters Perimeter = 38 meters Substitute the known values into the formula.
   Full step-by-step solution

   Let's solve this step by step. --- **Step 1: Recall the perimeter formula for a rectangle.** The perimeter \( P \) of a rectangle is given by: P = 2 × (length + width) --- **Step 2: Write down the given values.** Length = 12 meters Perimeter = 38 meters --- **Step 3: Substitute the known values into the formula.** 38 = 2 × (12 + width) --- **Step 4: Divide both sides by 2 to isolate the length + width term.** 38 ÷ 2 = 12 + width 19 = 12 + width --- **Step 5: Subtract 12 from both sides to solve for width.** 19 − 12 = width 7 = width --- **Step 6: Conclusion** The width of the garden is **7 meters**.

8. Noah is planning a school trip and needs to calculate transportation costs. The bus company charges a flat fee of $180 plus $12 per student. If Noah's total budget for transportation is $1,500, how many students can go on the trip? Answer: 110 Solution: The total cost is the flat fee plus the cost per student: 180 + 12x Set this equal to the budget: 180 + 12x = 1500 Subtract 180 from both sides: 12x = 1320 Divide both sides by 12: x = 110 Check: 180 + 12(110) = 180 + 1320 = 1500 The answer is 110 students.
   Full step-by-step solution

   Step 1: Let x represent the number of students Step 2: The total cost is the flat fee plus the cost per student: 180 + 12x Step 3: Set this equal to the budget: 180 + 12x = 1500 Step 4: Subtract 180 from both sides: 12x = 1320 Step 5: Divide both sides by 12: x = 110 Step 6: Check: 180 + 12(110) = 180 + 1320 = 1500 The answer is 110 students.

9. 3(x + 7) - 5 = 34 Answer: 6 Solution: 3(x + 7) - 5 = 34 Add 5 to both sides to remove the -5 from the left side. 3(x + 7) - 5 + 5 = 34 + 5 3(x + 7) = 39 Divide both sides by 3 to isolate the term (x + 7).
   Full step-by-step solution

   Let's solve the equation step-by-step. We start with: 3(x + 7) - 5 = 34 **Step 1:** Add 5 to both sides to remove the -5 from the left side. 3(x + 7) - 5 + 5 = 34 + 5 This simplifies to: 3(x + 7) = 39 **Step 2:** Divide both sides by 3 to isolate the term (x + 7). [3(x + 7)] / 3 = 39 / 3 This simplifies to: x + 7 = 13 **Step 3:** Subtract 7 from both sides to solve for x. x + 7 - 7 = 13 - 7 This gives: x = 6 **Final check:** Substitute x = 6 into the original equation: 3(6 + 7) - 5 = 3(13) - 5 = 39 - 5 = 34. It matches the original equation, so the solution is correct. **Answer:** x = 6