Linear Equations

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Answer Key & Explanations

Linear Equations · Grade 8 · Worksheet 3

1. 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 is the width of Liam's garden? Answer: 6 Solution: Let’s go step by step. Let the width of the garden be \( w \) meters. The length is 5 meters more than twice the width, so: length \( l = 2w + 5 \).
   Full step-by-step solution

   Let’s go 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: \( P = 2 \times (\text{length} + \text{width}) \). Given \( P = 46 \), we have: \( 2 \times (l + w) = 46 \). --- **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: Divide both sides by 2** \( 3w + 5 = 23 \). --- **Step 6: Solve for \( w \)** \( 3w = 23 - 5 \) \( 3w = 18 \) \( w = 6 \). --- **Step 7: Interpret the result** The width of Liam’s garden is 6 meters. --- **Final answer:** 6

2. Kaia is planning a rectangular garden in her backyard. The length of the garden is 7 feet more than three times its width. If the perimeter of the garden is 78 feet, what is the width of the garden in feet? Answer: 8 Solution: Let w represent the width of the garden in feet. The length is 7 feet more than three times the width, so length = 3w + 7. The perimeter of a rectangle is given by P = 2(length + width).
   Full step-by-step solution

   Step 1: Let w represent the width of the garden in feet. Step 2: The length is 7 feet more than three times the width, so length = 3w + 7. Step 3: The perimeter of a rectangle is given by P = 2(length + width). Step 4: Substitute the given perimeter and expressions: 78 = 2((3w + 7) + w). Step 5: Simplify inside the parentheses: 78 = 2(4w + 7). Step 6: Divide both sides by 2: 39 = 4w + 7. Step 7: Subtract 7 from both sides: 32 = 4w. Step 8: Divide both sides by 4: w = 8. The width of the garden is 8 feet.

3. Maya is planning a road trip and needs to calculate her total driving time. She will drive 240 miles at an average speed of 60 miles per hour, then stop for a 45-minute lunch break. After lunch, she will drive another 150 miles at 50 miles per hour. If she starts her trip at 8:00 AM, at what time will she reach her destination? Answer: 3:15 PM Solution: Calculate time for first driving segment: 240 miles ÷ 60 mph = 4 hours Calculate time for second driving segment: 150 miles ÷ 50 mph = 3 hours Convert lunch break to hours: 45 minutes = 45/60 = 0.75 hours Calculate total trip duration: 4 hours + 0.75 hours + 3 hours = 7.75 hours Convert 7.75…
   Full step-by-step solution

   Step 1: Calculate time for first driving segment: 240 miles ÷ 60 mph = 4 hours Step 2: Calculate time for second driving segment: 150 miles ÷ 50 mph = 3 hours Step 3: Convert lunch break to hours: 45 minutes = 45/60 = 0.75 hours Step 4: Calculate total trip duration: 4 hours + 0.75 hours + 3 hours = 7.75 hours Step 5: Convert 7.75 hours to hours and minutes: 7 hours + 0.75 × 60 minutes = 7 hours 45 minutes Step 6: Add duration to start time: 8:00 AM + 7 hours = 3:00 PM, then add 45 minutes = 3:45 PM The answer is 3:45 PM.

4. 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

5. Emma is planning a road trip and needs to calculate how much gas she'll need. Her car can travel 28 miles per gallon on the highway. The total distance of her trip is 350 miles. If gas costs $3.50 per gallon, how much will Emma spend on gas for the entire trip? Answer: 43.75 Solution: Calculate how many gallons of gas Emma needs for the trip. Total distance ÷ Miles per gallon = 350 ÷ 28 = 12.5 gallons Calculate the total cost of gas.
   Full step-by-step solution

   Step 1: Calculate how many gallons of gas Emma needs for the trip. Total distance ÷ Miles per gallon = 350 ÷ 28 = 12.5 gallons Step 2: Calculate the total cost of gas. Gallons needed × Price per gallon = 12.5 × 3.50 = 43.75 Step 3: The total cost for gas is $43.75.

6. Maya is planning a road trip and needs to rent a car. The rental company charges a base fee of $35 plus $0.25 per mile driven. If Maya has budgeted $120 for the car rental, how many miles can she drive without exceeding her budget? Answer: 340 Solution: Let m represent the number of miles Maya can drive. The total cost is the base fee plus the per-mile charge: 35 + 0.25m Set this equal to her budget: 35 + 0.25m = 120 Subtract 35 from both sides: 0.25m = 85 Divide both sides by 0.25: m = 85 ÷ 0.25 85 ÷ 0.25 = 340 The answer is 340 miles.
   Full step-by-step solution

   Step 1: Let m represent the number of miles Maya can drive. Step 2: The total cost is the base fee plus the per-mile charge: 35 + 0.25m Step 3: Set this equal to her budget: 35 + 0.25m = 120 Step 4: Subtract 35 from both sides: 0.25m = 85 Step 5: Divide both sides by 0.25: m = 85 ÷ 0.25 Step 6: 85 ÷ 0.25 = 340 The answer is 340 miles.

7. Liam is designing a rectangular garden with a length that is 3 meters more than twice its width. The perimeter of the garden is 36 meters. What is the width of Liam's garden? Answer: 5 Solution: Let the width of the garden be \( w \) meters. The length is 3 meters more than twice the width, so: length \( l = 2w + 3 \). \( P = 2 \times (l + w) \).
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Define variables** Let the width of the garden be \( w \) meters. The length is 3 meters more than twice the width, so: length \( l = 2w + 3 \). --- **Step 2: Write the perimeter formula** The perimeter \( P \) of a rectangle is: \( P = 2 \times (l + w) \). We are told \( P = 36 \), so: \( 2 \times (l + w) = 36 \). --- **Step 3: Substitute the expression for length** Substitute \( l = 2w + 3 \) into the perimeter equation: \( 2 \times ( (2w + 3) + w ) = 36 \). --- **Step 4: Simplify inside the parentheses** Inside: \( 2w + 3 + w = 3w + 3 \). So: \( 2 \times (3w + 3) = 36 \). --- **Step 5: Solve for \( w \)** Divide both sides by 2: \( 3w + 3 = 18 \). Subtract 3 from both sides: \( 3w = 15 \). Divide by 3: \( w = 5 \). --- **Step 6: Interpret the result** The width is 5 meters. --- **Final answer:** 5

8. Charlotte is comparing two different cell phone plans. Plan A charges a monthly fee of $27 plus $0.12 per text message. Plan B charges a monthly fee of $42 plus $0.07 per text message. How many text messages would Charlotte need to send in a month for the two plans to cost the same amount? Answer: 300 Solution: Let x represent the number of text messages. Write the cost for Plan A: 27 + 0.12x Write the cost for Plan B: 42 + 0.07x Set the costs equal: 27 + 0.12x = 42 + 0.07x Subtract 0.07x from both sides: 27 + 0.05x = 42 Subtract 27 from both sides: 0.05x = 15 Divide both sides by 0.05: x = 15 ÷ 0.05 =…
   Full step-by-step solution

   Step 1: Let x represent the number of text messages. Step 2: Write the cost for Plan A: 27 + 0.12x Step 3: Write the cost for Plan B: 42 + 0.07x Step 4: Set the costs equal: 27 + 0.12x = 42 + 0.07x Step 5: Subtract 0.07x from both sides: 27 + 0.05x = 42 Step 6: Subtract 27 from both sides: 0.05x = 15 Step 7: Divide both sides by 0.05: x = 15 ÷ 0.05 = 300 The answer is 300 text messages.