One Variable Equations

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One Variable Equations · Grade 6 · Worksheet 3

1. x + 2048 = 5000. Solve for x. Answer: 2952 Solution: The equation is x + 2048 = 5000. To isolate x, subtract 2048 from both sides of the equation. x + 2048 - 2048 = 5000 - 2048 Simplify: x = 2952 Check: 2952 + 2048 = 5000 ✓ The answer is 2952.
   Full step-by-step solution

   Step 1: The equation is x + 2048 = 5000. Step 2: To isolate x, subtract 2048 from both sides of the equation. Step 3: x + 2048 - 2048 = 5000 - 2048 Step 4: Simplify: x = 2952 Step 5: Check: 2952 + 2048 = 5000 ✓ The answer is 2952.

2. 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**.

3. Noah is designing a community garden for his school. The garden will be rectangular, with a length that is 8 meters more than twice its width. If the total perimeter of the garden needs to be 76 meters to fit within the available space, what is the width of the garden? Answer: 10 Solution: Let w represent the width of the garden in meters. The length is 8 meters more than twice the width, so length = 2w + 8 The perimeter formula for a rectangle is P = 2 × length + 2 × width Substitute the known values: 76 = 2(2w + 8) + 2(w) Expand the equation: 76 = 4w + 16 + 2w Combine like…
   Full step-by-step solution

   Step 1: Let w represent the width of the garden in meters. Step 2: The length is 8 meters more than twice the width, so length = 2w + 8 Step 3: The perimeter formula for a rectangle is P = 2 × length + 2 × width Step 4: Substitute the known values: 76 = 2(2w + 8) + 2(w) Step 5: Expand the equation: 76 = 4w + 16 + 2w Step 6: Combine like terms: 76 = 6w + 16 Step 7: Subtract 16 from both sides: 60 = 6w Step 8: Divide both sides by 6: w = 10 Step 9: The width of the garden is 10 meters.

4. Maya is saving money to buy a new video game. The game costs $78. Maya already has $40. How much more money does Maya need to save to buy the game? Answer: 38 Solution: Let x represent the additional money needed. Write the equation: 40 + x = 78. Subtract 40 from both sides: x = 78 - 40.
   Full step-by-step solution

   Step 1: Let x represent the additional money needed. Step 2: Write the equation: 40 + x = 78. Step 3: Subtract 40 from both sides: x = 78 - 40. Step 4: Calculate: x = 38. Maya needs to save $38 more.

5. 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.

6. Olivia is saving money to buy a new video game. The game costs $44. Olivia already has $25. How much more money does Olivia need to save to buy the game? Answer: 19 Solution: Let x represent the additional money needed. Write the equation: 25 + x = 44. Subtract 25 from both sides: x = 44 - 25.
   Full step-by-step solution

   Step 1: Let x represent the additional money needed. Step 2: Write the equation: 25 + x = 44. Step 3: Subtract 25 from both sides: x = 44 - 25. Step 4: Calculate: x = 19. Olivia needs to save $19 more.

7. 3(2x - 7) + 15 = 42 Answer: 8 Solution: Distribute the 3: 3 × 2x = 6x and 3 × (-7) = -21 Rewrite the equation: 6x - 21 + 15 = 42 Combine like terms: 6x - 6 = 42 Add 6 to both sides: 6x = 48 Divide both sides by 6: x = 8 Check: 3(2×8 - 7) + 15 = 3(16 - 7) + 15 = 3×9 + 15 = 27 + 15 = 42 The answer is 8.
   Full step-by-step solution

   Step 1: Distribute the 3: 3 × 2x = 6x and 3 × (-7) = -21 Step 2: Rewrite the equation: 6x - 21 + 15 = 42 Step 3: Combine like terms: 6x - 6 = 42 Step 4: Add 6 to both sides: 6x = 48 Step 5: Divide both sides by 6: x = 8 Step 6: Check: 3(2×8 - 7) + 15 = 3(16 - 7) + 15 = 3×9 + 15 = 27 + 15 = 42 The answer is 8.

8. Noah is looking at a scale drawing of a rectangular playground. The length of the playground in the drawing is represented by the variable x. The width is 7 meters less than three times the length. If the width is 29 meters, what is the length of the playground in meters? Write an equation and solve for x. Answer: 12 Solution: Write the equation from the description: 3x - 7 = 29. To isolate the term with x, add 7 to both sides of the equation: 3x - 7 + 7 = 29 + 7, which simplifies to 3x = 36.
   Full step-by-step solution

   Step 1: Write the equation from the description: 3x - 7 = 29. Step 2: To isolate the term with x, add 7 to both sides of the equation: 3x - 7 + 7 = 29 + 7, which simplifies to 3x = 36. Step 3: Divide both sides by 3 to solve for x: 3x / 3 = 36 / 3, so x = 12. Therefore, the length of the playground is 12 meters.