Linear Equations

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Answer Key & Explanations

Linear Equations · Grade 8 · Worksheet 3

1. Liam is saving money to buy a new video game that costs $65. He already has some money saved and earns $12 per week from his part-time job. After 4 weeks of saving, he has exactly enough to buy the game. Write and solve an equation to find how much money Liam had saved initially. Answer: 17 Solution: Let’s break this down step by step. Let \( x \) = the amount of money Liam had saved initially (in dollars). Liam earns $12 per week from his job.
   Full step-by-step solution

   Let’s break this down step by step. --- **Step 1: Define a variable for the initial savings** Let \( x \) = the amount of money Liam had saved initially (in dollars). --- **Step 2: Understand the weekly earnings** Liam earns $12 per week from his job. After 4 weeks, the total money earned from the job is: \( 12 \times 4 = 48 \) dollars. --- **Step 3: Write the total money after 4 weeks** Total money after 4 weeks = initial savings + money earned from job That is: \( x + 48 \) --- **Step 4: Set up the equation** We know that after 4 weeks, he has exactly enough to buy the game, which costs $65. So: \( x + 48 = 65 \) --- **Step 5: Solve for \( x \)** Subtract 48 from both sides: \( x = 65 - 48 \) \( x = 17 \) --- **Step 6: Interpret the result** Liam initially had $17 saved. --- **Final answer:** 17

2. Aisha is designing a rectangular banner for the school science fair. The length of the banner is 3 feet less than three times its width. If the perimeter of the banner is 50 feet, what is the width of the banner in feet? Answer: 7 Solution: Let w represent the width of the banner in feet. The length is 3 feet less than three times the width, so length = 3w - 3. The perimeter formula for a rectangle is P = 2(length + width).
   Full step-by-step solution

   Step 1: Let w represent the width of the banner in feet. Step 2: The length is 3 feet less than three times the width, so length = 3w - 3. Step 3: The perimeter formula for a rectangle is P = 2(length + width). Step 4: Substitute the known values: 50 = 2((3w - 3) + w) Step 5: Simplify inside the parentheses: 50 = 2(4w - 3) Step 6: Distribute the 2: 50 = 8w - 6 Step 7: Add 6 to both sides: 56 = 8w Step 8: Divide both sides by 8: w = 7 Step 9: The width of the banner is 7 feet.

3. Isabella is comparing two different tutoring service plans. Plan A charges a flat registration fee of $42 and then $27 per hour of tutoring. Plan B charges a flat registration fee of $72 and then $22 per hour of tutoring. How many hours of tutoring would Isabella need to take for the total cost of both plans to be the same? Answer: 6 Solution: Let h be the number of hours of tutoring. Cost for Plan A = 42 + 27h Cost for Plan B = 72 + 22h Set the costs equal: 42 + 27h = 72 + 22h Subtract 22h from both sides: 42 + 5h = 72 Subtract 42 from both sides: 5h = 30 Divide both sides by 5: h = 6 The answer is 6 hours.
   Full step-by-step solution

   Step 1: Let h be the number of hours of tutoring. Step 2: Cost for Plan A = 42 + 27h Step 3: Cost for Plan B = 72 + 22h Step 4: Set the costs equal: 42 + 27h = 72 + 22h Step 5: Subtract 22h from both sides: 42 + 5h = 72 Step 6: Subtract 42 from both sides: 5h = 30 Step 7: Divide both sides by 5: h = 6 The answer is 6 hours.

4. Emma is drawing a rectangle on a coordinate grid. The vertices of the rectangle are at (0, 0), (x, 0), (x, 10), and (0, 10). She then draws a diagonal line from the bottom-left corner (0, 0) to the top-right corner (x, 10). The slope of this diagonal line is 2.5. What is the value of x? Answer: 4 Solution: The slope of a line between two points (x1, y1) and (x2, y2) is (y2 - y1) / (x2 - x1). Here, the points are (0, 0) and (x, 10). So, slope = (10 - 0) / (x - 0) = 10 / x.
   Full step-by-step solution

   Step 1: The slope of a line between two points (x1, y1) and (x2, y2) is (y2 - y1) / (x2 - x1). Step 2: Here, the points are (0, 0) and (x, 10). So, slope = (10 - 0) / (x - 0) = 10 / x. Step 3: The slope is given as 2.5, so set up the equation: 10 / x = 2.5. Step 4: Multiply both sides by x: 10 = 2.5x. Step 5: Divide both sides by 2.5: x = 10 / 2.5 = 4. The answer is 4.

5. Emma is comparing two different subscription boxes for art supplies. Box A costs a flat fee of $13 per month plus $7 per additional craft kit. Box B costs a flat fee of $19 per month plus $5 per additional craft kit. How many additional craft kits must Emma order in a month for the total costs of the two boxes to be equal? Answer: 3 Solution: Let x represent the number of additional craft kits ordered in a month. Total cost for Box A = 13 + 7x Total cost for Box B = 19 + 5x Set the costs equal: 13 + 7x = 19 + 5x Subtract 5x from both sides: 13 + 7x - 5x = 19 - 5x + 5x, so 13 + 2x = 19 Subtract 13 from both sides: 13 - 13 + 2x = 19 -…
   Full step-by-step solution

   Step 1: Let x represent the number of additional craft kits ordered in a month. Step 2: Total cost for Box A = 13 + 7x Step 3: Total cost for Box B = 19 + 5x Step 4: Set the costs equal: 13 + 7x = 19 + 5x Step 5: Subtract 5x from both sides: 13 + 7x - 5x = 19 - 5x + 5x, so 13 + 2x = 19 Step 6: Subtract 13 from both sides: 13 - 13 + 2x = 19 - 13, so 2x = 6 Step 7: Divide both sides by 2: x = 3 The answer is 3 additional craft kits.

6. Emma is planning a road trip and needs to calculate her fuel costs. Her car's fuel efficiency is 28 miles per gallon, and she plans to drive 350 miles. If gasoline costs $3.25 per gallon, how much will Emma spend on gasoline for her trip? Round your answer to the nearest cent. Answer: 40.63 Solution: Distance = 350 miles Fuel efficiency = 28 miles per gallon Gallons needed = 350 ÷ 28 = 12.5 gallons Cost per gallon = $3.25 Total cost = 12.5 × 3.25 = $40.625 $40.625 rounds to $40.63 The answer is $40.63.
   Full step-by-step solution

   Step 1: Calculate how many gallons of gasoline Emma needs Distance = 350 miles Fuel efficiency = 28 miles per gallon Gallons needed = 350 ÷ 28 = 12.5 gallons Step 2: Calculate the total cost Cost per gallon = $3.25 Total cost = 12.5 × 3.25 = $40.625 Step 3: Round to the nearest cent $40.625 rounds to $40.63 The answer is $40.63.

7. Olivia is drawing a square on a coordinate grid. The square has vertices at (1, 3), (1, y), (x, y), and (x, 3). The perimeter of the square is 28 units. What is the value of x? Answer: 8 Solution: The perimeter of a square is 4 times the side length. Let s be the side length. Then 4s = 28.
   Full step-by-step solution

   Step 1: The perimeter of a square is 4 times the side length. Let s be the side length. Then 4s = 28. Step 2: Divide both sides by 4: s = 7. Step 3: The square has vertices (1, 3) and (1, y) sharing the same x-coordinate, so the vertical side length is y - 3. Since the side length is 7, y - 3 = 7, so y = 10. Step 4: The square also has vertices (1, 3) and (x, 3) sharing the same y-coordinate, so the horizontal side length is x - 1. Since the side length is 7, x - 1 = 7. Step 5: Add 1 to both sides: x = 8. The answer is 8.