Linear Equations

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Linear Equations · Grade 8 · Worksheet 1

1. Noah is comparing two different gym membership plans. Plan A charges a flat fee of $35 per month plus $8 per visit. Plan B charges a flat fee of $15 per month plus $13 per visit. After how many visits in a month will the total cost of both plans be the same? Answer: 4 Solution: Let x be the number of visits in a month. Cost for Plan A = 35 + 8x Cost for Plan B = 15 + 13x Set the costs equal: 35 + 8x = 15 + 13x Subtract 8x from both sides: 35 = 15 + 5x Subtract 15 from both sides: 20 = 5x Divide both sides by 5: x = 4 The answer is 4 visits.
   Full step-by-step solution

   Step 1: Let x be the number of visits in a month. Step 2: Cost for Plan A = 35 + 8x Step 3: Cost for Plan B = 15 + 13x Step 4: Set the costs equal: 35 + 8x = 15 + 13x Step 5: Subtract 8x from both sides: 35 = 15 + 5x Step 6: Subtract 15 from both sides: 20 = 5x Step 7: Divide both sides by 5: x = 4 The answer is 4 visits.

2. Liam is designing a rectangular garden for his school project. The length of the garden is 5 feet more than twice its width. If the perimeter of the garden is 82 feet, what is the width of the garden in feet? Answer: 12 Solution: Let \( w \) = width of the garden (in feet). Let \( l \) = length of the garden (in feet).
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Define variables** Let \( w \) = width of the garden (in feet). Let \( l \) = length of the garden (in feet). --- **Step 2: Translate the problem into equations** The problem says: "Length is 5 feet more than twice its width" So: \( l = 2w + 5 \) The perimeter of a rectangle is: \( P = 2l + 2w \) Given \( P = 82 \), so: \( 2l + 2w = 82 \) --- **Step 3: Substitute the expression for \( l \) into the perimeter equation** From \( l = 2w + 5 \), substitute into \( 2l + 2w = 82 \): \( 2(2w + 5) + 2w = 82 \) --- **Step 4: Simplify and solve for \( w \)** First, distribute: \( 4w + 10 + 2w = 82 \) Combine like terms: \( 6w + 10 = 82 \) Subtract 10 from both sides: \( 6w = 72 \) Divide both sides by 6: \( w = 12 \) --- **Step 5: Interpret the result** The width \( w = 12 \) feet. --- **Final answer:** 12

3. Liam is designing a rectangular garden with a length that is 5 meters more than twice its width. The perimeter of the garden is 46 meters. What are the dimensions of Liam's garden? Answer: width = 6 meters, length = 17 meters Solution: Let the width of the garden be \( w \) meters. The length is 5 meters more than twice the width, so: length \( l = 2w + 5 \). \( P = 2 \times (\text{length} + \text{width}) \).
   Full step-by-step solution

   Let's solve step by step. --- **Step 1: Define variables** Let the width of the garden be \( w \) meters. The length is 5 meters more than twice the width, so: length \( l = 2w + 5 \). --- **Step 2: Write the perimeter formula** The perimeter \( P \) of a rectangle is: \( P = 2 \times (\text{length} + \text{width}) \). We are told \( P = 46 \), so: \( 2 \times (l + w) = 46 \). --- **Step 3: Substitute the expression for length** Substitute \( l = 2w + 5 \) into the perimeter equation: \( 2 \times ( (2w + 5) + w ) = 46 \). --- **Step 4: Simplify inside the parentheses** \( (2w + 5) + w = 3w + 5 \). So: \( 2 \times (3w + 5) = 46 \). --- **Step 5: Solve for \( w \)** Divide both sides by 2: \( 3w + 5 = 23 \). Subtract 5 from both sides: \( 3w = 18 \). Divide by 3: \( w = 6 \). --- **Step 6: Find the length** \( l = 2w + 5 = 2 \times 6 + 5 = 12 + 5 = 17 \). --- **Step 7: State the answer** Width = 6 meters, Length = 17 meters. --- **Final check:** Perimeter = \( 2 \times (17 + 6) = 2 \times 23 = 46 \), correct. Length is 5 more than twice the width: \( 2 \times 6 + 5 = 17 \), correct.

4. Mason is comparing two different dog-walking jobs. For Job A, he earns $7 per dog walked plus a $27 daily bonus. For Job B, he earns $12 per dog walked plus a $17 daily bonus. How many dogs does Mason need to walk in a day for both jobs to pay the same amount? Answer: 2 Solution: Let d represent the number of dogs walked in a day. Job A pay = 7d + 27 Job B pay = 12d + 17 7d + 27 = 12d + 17 Subtract 7d from both sides: 7d - 7d + 27 = 12d - 7d + 17 27 = 5d + 17 Subtract 17 from both sides: 27 - 17 = 5d + 17 - 17 10 = 5d Divide both sides by 5: 10/5 = 5d/5 2 = d Mason must…
   Full step-by-step solution

   Let d represent the number of dogs walked in a day. Job A pay = 7d + 27 Job B pay = 12d + 17 Set the two expressions equal: 7d + 27 = 12d + 17 Subtract 7d from both sides: 7d - 7d + 27 = 12d - 7d + 17 27 = 5d + 17 Subtract 17 from both sides: 27 - 17 = 5d + 17 - 17 10 = 5d Divide both sides by 5: 10/5 = 5d/5 2 = d Mason must walk 2 dogs in a day for both jobs to pay the same amount. The answer is 2.

5. Liam is saving money to buy a new video game that costs $65. He already has $20 saved from his allowance. He plans to save the same amount each week from his part-time job. After 5 weeks of saving, he has exactly enough money to buy the game. How much money does Liam save each week? Answer: 9 Solution: - Cost of video game = $65 - Money Liam already has = $20 - He saves the same amount each week for 5 weeks. - After 5 weeks, total money = $65 (exactly enough to buy the game). Let \( x \) = amount saved each week.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understand the problem** - Cost of video game = $65 - Money Liam already has = $20 - He saves the same amount each week for 5 weeks. - After 5 weeks, total money = $65 (exactly enough to buy the game). --- **Step 2: Set up the relationship** Let \( x \) = amount saved each week. After 5 weeks, the money saved from the job = \( 5 \times x \). Total money after 5 weeks = initial money + money saved from job \[ 20 + 5x = 65 \] --- **Step 3: Solve for \( x \)** Subtract 20 from both sides: \[ 5x = 65 - 20 \] \[ 5x = 45 \] Divide both sides by 5: \[ x = 45 / 5 \] \[ x = 9 \] --- **Step 4: Interpret the result** Liam saves $9 each week. --- **Step 5: Verify** Initial money = $20 After 5 weeks of saving $9 each week: Money from job = \( 5 \times 9 = 45 \) Total = \( 20 + 45 = 65 \) ✔ --- **Final answer:** 9

6. 2(3x - 5) + 7 = 4(x + 1) - 3 = ? Answer: 2 Solution: Distribute on both sides: 2(3x - 5) + 7 = 6x - 10 + 7 and 4(x + 1) - 3 = 4x + 4 - 3 Combine like terms on both sides: 6x - 3 = 4x + 1 Subtract 4x from both sides: 6x - 4x - 3 = 1 → 2x - 3 = 1 Add 3 to both sides: 2x = 4 Divide both sides by 2: x = 2 The answer is 2.
   Full step-by-step solution

   Step 1: Distribute on both sides: 2(3x - 5) + 7 = 6x - 10 + 7 and 4(x + 1) - 3 = 4x + 4 - 3 Step 2: Combine like terms on both sides: 6x - 3 = 4x + 1 Step 3: Subtract 4x from both sides: 6x - 4x - 3 = 1 → 2x - 3 = 1 Step 4: Add 3 to both sides: 2x = 4 Step 5: Divide both sides by 2: x = 2 The answer is 2.

7. Emma is planning a road trip and needs to calculate her fuel costs. Her car can travel 320 miles on a full tank of gas. If gas costs $3.75 per gallon and her car's fuel efficiency is 25 miles per gallon, how much will it cost Emma to fill up her empty gas tank? Answer: 48 Solution: First, find how many gallons Emma's gas tank holds. The car travels 320 miles on a full tank and gets 25 miles per gallon.
   Full step-by-step solution

   Step 1: First, find how many gallons Emma's gas tank holds. The car travels 320 miles on a full tank and gets 25 miles per gallon. Step 2: Calculate gallons in a full tank: 320 miles ÷ 25 miles/gallon = 12.8 gallons Step 3: Now calculate the cost to fill the empty tank. Gas costs $3.75 per gallon. Step 4: Multiply gallons by cost per gallon: 12.8 gallons × $3.75/gallon = $48.00 The answer is 48.

8. Aisha is planning a road trip and needs to calculate how much gas her car will use. Her car's fuel efficiency is 28 miles per gallon on the highway. The total distance of her trip is 392 miles, and gas costs $3.50 per gallon. If Aisha wants to know how much she'll spend on gas for the entire trip, which equation should she solve first to find the number of gallons needed? Answer: 28g = 392 Solution: When solving problems involving rates like fuel efficiency, it's helpful to set up an equation where the rate multiplied by the quantity equals the total.
   Full step-by-step solution

   When solving problems involving rates like fuel efficiency, it's helpful to set up an equation where the rate multiplied by the quantity equals the total. For example, if you know how many items you can buy with one dollar and your total budget, you can find how many dollars you need.