One Variable Equations

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One Variable Equations · Grade 6 · Worksheet 2

1. x + 2845 = 5723. Solve for x. Answer: 2878 Solution: The equation is x + 2845 = 5723. To isolate x, subtract 2845 from both sides of the equation. x + 2845 - 2845 = 5723 - 2845 Simplify: x = 2878 Check: 2878 + 2845 = 5723 ✓ The answer is 2878.
   Full step-by-step solution

   Step 1: The equation is x + 2845 = 5723. Step 2: To isolate x, subtract 2845 from both sides of the equation. Step 3: x + 2845 - 2845 = 5723 - 2845 Step 4: Simplify: x = 2878 Step 5: Check: 2878 + 2845 = 5723 ✓ The answer is 2878.

2. x + 2847 = 5723. Solve for x. Answer: 2876 Solution: The equation is x + 2847 = 5723. To isolate x, subtract 2847 from both sides of the equation. x + 2847 - 2847 = 5723 - 2847 Simplify: x = 2876 Check: 2876 + 2847 = 5723.
   Full step-by-step solution

   Step 1: The equation is x + 2847 = 5723. Step 2: To isolate x, subtract 2847 from both sides of the equation. Step 3: x + 2847 - 2847 = 5723 - 2847 Step 4: Simplify: x = 2876 Step 5: Check: 2876 + 2847 = 5723. This is correct. The answer is 2876.

3. A rectangular prism is drawn with dimensions: length = 15 cm, width = 10 cm, and height = 8 cm. If you were to draw a line connecting the bottom front left corner to the top back right corner (the space diagonal), what is the length of this diagonal line in centimeters? Round your answer to the nearest tenth. Answer: 19.7 Solution: To find the space diagonal of a rectangular prism, use the 3D Pythagorean theorem: diagonal = sqrt(length² + width² + height²) Square each dimension: 15² = 225, 10² = 100, 8² = 64 Add the squares: 225 + 100 + 64 = 389 Take the square root: sqrt(389) ≈ 19.723 Round to the nearest tenth: 19.7…
   Full step-by-step solution

   Step 1: To find the space diagonal of a rectangular prism, use the 3D Pythagorean theorem: diagonal = sqrt(length² + width² + height²) Step 2: Square each dimension: 15² = 225, 10² = 100, 8² = 64 Step 3: Add the squares: 225 + 100 + 64 = 389 Step 4: Take the square root: sqrt(389) ≈ 19.723 Step 5: Round to the nearest tenth: 19.7 Therefore, the length of the space diagonal is 19.7 cm.

4. x + 1272 = 2847 Answer: 1575 Solution: The equation is x + 1272 = 2847. To isolate x, subtract 1272 from both sides of the equation. x + 1272 - 1272 = 2847 - 1272 Simplify: x = 1575 Check: 1575 + 1272 = 2847 ✓ The answer is 1575.
   Full step-by-step solution

   Step 1: The equation is x + 1272 = 2847. Step 2: To isolate x, subtract 1272 from both sides of the equation. Step 3: x + 1272 - 1272 = 2847 - 1272 Step 4: Simplify: x = 1575 Step 5: Check: 1575 + 1272 = 2847 ✓ The answer is 1575.

5. 3(2x - 5) + 7 = 34 Answer: 7 Solution: Distribute the 3 to both terms inside the parentheses: 3 × 2x = 6x and 3 × (-5) = -15 Rewrite the equation: 6x - 15 + 7 = 34 Combine the constant terms: -15 + 7 = -8 The equation becomes: 6x - 8 = 34 Add 8 to both sides: 6x - 8 + 8 = 34 + 8 Simplify: 6x = 42 Divide both sides by 6: 6x ÷ 6 = 42 ÷…
   Full step-by-step solution

   Step 1: Distribute the 3 to both terms inside the parentheses: 3 × 2x = 6x and 3 × (-5) = -15 Step 2: Rewrite the equation: 6x - 15 + 7 = 34 Step 3: Combine the constant terms: -15 + 7 = -8 Step 4: The equation becomes: 6x - 8 = 34 Step 5: Add 8 to both sides: 6x - 8 + 8 = 34 + 8 Step 6: Simplify: 6x = 42 Step 7: Divide both sides by 6: 6x ÷ 6 = 42 ÷ 6 Step 8: Simplify: x = 7 The answer is 7.

6. Liam is saving money to buy a new video game that costs $65. He already has $28 saved from his allowance. For his birthday, his grandparents give him some money, and after adding this to his savings, he has exactly enough to buy the game. Which equation could be used to find m, the amount of money Liam received for his birthday? Answer: 28 + m = 65 Solution: Understand what we know. - The video game costs $65. - Liam already has $28 saved.
   Full step-by-step solution

   Let's go step by step. --- **Step 1: Understand what we know.** - The video game costs $65. - Liam already has $28 saved. - He gets some birthday money from his grandparents. Let’s call this amount **m**. - After adding the birthday money to his savings, he has exactly enough to buy the game. --- **Step 2: Set up the relationship.** The total money Liam has after his birthday is: Money he already had + Birthday money = Cost of the game That is: 28 + m = 65 --- **Step 3: Explain why this equation works.** We want to find **m**, the birthday money. If he had $28 and then got $m more, the total is 28 + m. We know this total equals $65 because the problem says he then has exactly enough to buy the game. So the equation is: 28 + m = 65 --- **Step 4: Final check.** No other numbers are involved, so no extra steps are needed. The equation that represents the situation is: 28 + m = 65 --- **Final answer:** 28 + m = 65

7. Noah is designing a scale model of a solar system for his science project. The actual distance from Earth to Mars is approximately 225 million kilometers. If Noah's model uses a scale where 1 centimeter represents 15 million kilometers, how many centimeters apart should he place the Earth and Mars models in his display? Answer: 15 Solution: Identify the actual distance: 225 million kilometers Identify the scale: 1 centimeter = 15 million kilometers Set up the equation: model distance = actual distance ÷ scale factor Calculate: 225 ÷ 15 = 15 The Earth and Mars models should be 15 centimeters apart in Noah's display.
   Full step-by-step solution

   Step 1: Identify the actual distance: 225 million kilometers Step 2: Identify the scale: 1 centimeter = 15 million kilometers Step 3: Set up the equation: model distance = actual distance ÷ scale factor Step 4: Calculate: 225 ÷ 15 = 15 Step 5: The Earth and Mars models should be 15 centimeters apart in Noah's display.

8. A rectangular prism is drawn with dimensions: length = 15 cm, width = 10 cm, and height = 8 cm. If you were to draw a line connecting the bottom front left corner to the top back right corner, what is the length of this diagonal line through the prism? Answer: sqrt(389) cm Solution: First find the diagonal of the base rectangle using the Pythagorean theorem. Base diagonal = sqrt(length^2 + width^2) = sqrt(15^2 + 10^2) = sqrt(225 + 100) = sqrt(325) Now use this base diagonal and the height to find the space diagonal through the prism.
   Full step-by-step solution

   Step 1: First find the diagonal of the base rectangle using the Pythagorean theorem. Base diagonal = sqrt(length^2 + width^2) = sqrt(15^2 + 10^2) = sqrt(225 + 100) = sqrt(325) Step 2: Now use this base diagonal and the height to find the space diagonal through the prism. Space diagonal = sqrt((base diagonal)^2 + height^2) = sqrt((sqrt(325))^2 + 8^2) = sqrt(325 + 64) = sqrt(389) The answer is sqrt(389) cm.