Linear Equations

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Linear Equations · Grade 8 · Worksheet 1

1. Mason is comparing two different job offers for the summer. At the first job, he will earn $27 per hour and receive a $42 signing bonus. At the second job, he will earn $22 per hour and receive a $72 signing bonus. For how many hours of work will the total earnings from both jobs be equal? Answer: 6 Solution: Let h represent the number of hours worked. Total earnings for first job: 27h + 42 Total earnings for second job: 22h + 72 Set them equal: 27h + 42 = 22h + 72 Subtract 22h from both sides: 27h - 22h + 42 = 72, so 5h + 42 = 72 Subtract 42 from both sides: 5h = 30 Divide both sides by 5: h = 6 The…
   Full step-by-step solution

   Step 1: Let h represent the number of hours worked. Step 2: Total earnings for first job: 27h + 42 Step 3: Total earnings for second job: 22h + 72 Step 4: Set them equal: 27h + 42 = 22h + 72 Step 5: Subtract 22h from both sides: 27h - 22h + 42 = 72, so 5h + 42 = 72 Step 6: Subtract 42 from both sides: 5h = 30 Step 7: Divide both sides by 5: h = 6 The answer is 6 hours.

2. Liam is designing a rectangular garden. The length of the garden is 3 meters more than twice its width. If the perimeter of the garden is 36 meters, what is the width of the garden in meters? 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** \( (2w + 3) + w = 3w + 3 \). So: \( 2 \times (3w + 3) = 36 \). --- **Step 5: Divide both sides by 2** \( 3w + 3 = 18 \). --- **Step 6: Subtract 3 from both sides** \( 3w = 15 \). --- **Step 7: Divide by 3** \( w = 5 \). --- **Step 8: Conclusion** The width of the garden is 5 meters. --- **Final check:** Width \( w = 5 \) m, length \( l = 2(5) + 3 = 13 \) m. Perimeter = \( 2 \times (5 + 13) = 2 \times 18 = 36 \) m. ✔ **Answer:** 5

3. 3(2x - 5) + 7 = 4(x + 3) - 2 = ? Answer: x = 6 Solution: 3(2x - 5) + 7 = 4(x + 3) - 2 Left side: 3(2x - 5) + 7 = 3 * 2x - 3 * 5 + 7 = 6x - 15 + 7 = 6x - 8 Right side: 4(x + 3) - 2 = 4 * x + 4 * 3 - 2 = 4x + 12 - 2 = 4x + 10 6x - 8 = 4x + 10 Subtract 4x from both sides: 6x - 8 - 4x = 4x + 10 - 4x 2x - 8 = 10 Add 8 to both sides: 2x - 8 + 8 = 10 + 8 2x…
   Full step-by-step solution

   Let's solve the equation step by step. We have: 3(2x - 5) + 7 = 4(x + 3) - 2 --- **Step 1: Expand both sides** Left side: 3(2x - 5) + 7 = 3 * 2x - 3 * 5 + 7 = 6x - 15 + 7 = 6x - 8 Right side: 4(x + 3) - 2 = 4 * x + 4 * 3 - 2 = 4x + 12 - 2 = 4x + 10 So the equation becomes: 6x - 8 = 4x + 10 --- **Step 2: Move x terms to one side** Subtract 4x from both sides: 6x - 8 - 4x = 4x + 10 - 4x 2x - 8 = 10 --- **Step 3: Move constant terms to the other side** Add 8 to both sides: 2x - 8 + 8 = 10 + 8 2x = 18 --- **Step 4: Solve for x** Divide both sides by 2: x = 18 / 2 x = 6 --- **Step 5: Check** Left side: 3(2*6 - 5) + 7 = 3(12 - 5) + 7 = 3*7 + 7 = 21 + 7 = 28 Right side: 4(6 + 3) - 2 = 4*9 - 2 = 36 - 2 = 28 Both sides equal, so the solution is correct. --- **Final answer:** x = 6

4. 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

5. A right triangle is drawn on a coordinate plane with vertices at (0,0), (12,0), and (0,5). A line parallel to the hypotenuse is drawn from the point (4,0) to intersect the vertical leg. This creates a smaller right triangle inside the larger one. What is the length of the vertical leg of this smaller triangle? Answer: 1.67 Solution: Identify the similar triangles. The small triangle formed by points (4,0), (0,0), and the intersection point on the vertical leg is similar to the large triangle formed by (12,0), (0,0), and (0,5).
   Full step-by-step solution

   Step 1: Identify the similar triangles. The small triangle formed by points (4,0), (0,0), and the intersection point on the vertical leg is similar to the large triangle formed by (12,0), (0,0), and (0,5). Step 2: Set up the proportion using corresponding sides. The horizontal side of the large triangle is 12 units, and the horizontal side of the small triangle is 4 units. Step 3: The ratio of similarity is 4/12 = 1/3. Step 4: Apply this ratio to the vertical leg of the large triangle, which is 5 units. Step 5: Calculate the vertical leg of the small triangle: (1/3) × 5 = 5/3 ≈ 1.67 Step 6: The length of the vertical leg of the smaller triangle is approximately 1.67 units.

6. 4(2x - 3) + 7 = 3(3x + 1) - 5 = ? Answer: -3 Solution: Left side: 4(2x - 3) = 8x - 12 Right side: 3(3x + 1) = 9x + 3 Equation becomes: 8x - 12 + 7 = 9x + 3 - 5 Left side: 8x - 5 Right side: 9x - 2 Equation becomes: 8x - 5 = 9x - 2 Subtract 9x from both sides: 8x - 5 - 9x = 9x - 2 - 9x Simplifies to: -x - 5 = -2 Add 5 to both sides: -x - 5 + 5 = -2 +…
   Full step-by-step solution

   Step 1: Distribute on both sides Left side: 4(2x - 3) = 8x - 12 Right side: 3(3x + 1) = 9x + 3 Equation becomes: 8x - 12 + 7 = 9x + 3 - 5 Step 2: Combine like terms on each side Left side: 8x - 5 Right side: 9x - 2 Equation becomes: 8x - 5 = 9x - 2 Step 3: Move variable terms to one side Subtract 9x from both sides: 8x - 5 - 9x = 9x - 2 - 9x Simplifies to: -x - 5 = -2 Step 4: Move constant terms to the other side Add 5 to both sides: -x - 5 + 5 = -2 + 5 Simplifies to: -x = 3 Step 5: Solve for x Multiply both sides by -1: -x × (-1) = 3 × (-1) x = -3 The answer is -3.

7. 7(2x - 3) + 12 = 3(5x + 2) - 17 = ? Answer: 2 Solution: Distribute on both sides. Left side: 7(2x - 3) + 12 = 14x - 21 + 12 Right side: 3(5x + 2) - 17 = 15x + 6 - 17 Combine like terms on each side.
   Full step-by-step solution

   Step 1: Distribute on both sides. Left side: 7(2x - 3) + 12 = 14x - 21 + 12 Right side: 3(5x + 2) - 17 = 15x + 6 - 17 Step 2: Combine like terms on each side. Left side: 14x - 9 Right side: 15x - 11 Equation: 14x - 9 = 15x - 11 Step 3: Move variable terms to one side. Subtract 14x from both sides. 14x - 9 - 14x = 15x - 11 - 14x -9 = x - 11 Step 4: Move constants. Add 11 to both sides. -9 + 11 = x - 11 + 11 2 = x Step 5: Write the solution. x = 2 The answer is 2.

8. Emma 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. The total distance of her trip is 392 miles. If gas costs $3.50 per gallon, how much will Emma spend on gas for the entire trip? Answer: 49 Solution: Calculate how many gallons of gas Emma needs for the trip. Total distance ÷ Fuel efficiency = Gallons needed 392 ÷ 28 = 14 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 ÷ Fuel efficiency = Gallons needed 392 ÷ 28 = 14 gallons Step 2: Calculate the total cost of gas. Gallons needed × Price per gallon = Total cost 14 × 3.50 = 49 The answer is $49.