Linear Equations

lessonbunny.com

Answer Key & Explanations

Linear Equations · Grade 8 · Worksheet 2

1. Liam is saving money to buy a new video game that costs $65. He already has $20 saved from his allowance. Each week, he earns an additional $15 by doing chores. Write and solve an equation to determine how many weeks it will take for Liam to have enough money to buy the video game. Answer: 3 Solution: Liam needs $65 total. He already has $20 saved. Each week he earns $15 from chores.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understand the problem** Liam needs $65 total. He already has $20 saved. Each week he earns $15 from chores. We want to find how many weeks (call it \( w \)) until he has enough. --- **Step 2: Set up the equation** Money after \( w \) weeks = starting money + weekly earnings × number of weeks \[ 20 + 15 \times w = 65 \] --- **Step 3: Solve for \( w \)** Subtract 20 from both sides: \[ 15w = 65 - 20 \] \[ 15w = 45 \] Divide both sides by 15: \[ w = 45 / 15 \] \[ w = 3 \] --- **Step 4: Interpret the result** It will take 3 weeks for Liam to have enough money. --- **Final answer:** 3

2. 3(2x - 5) + 7 = 4(x + 3) - 2 Answer: x = 7 Solution: To solve linear equations with variables on both sides, first use the distributive property to eliminate parentheses. Then combine like terms on each side.
   Full step-by-step solution

   To solve linear equations with variables on both sides, first use the distributive property to eliminate parentheses. Then combine like terms on each side. Finally, use inverse operations to isolate the variable term on one side and the constant terms on the other side.

3. Kaia is comparing two different savings accounts. Account A earns $9 in interest per month plus a one-time bonus of $35. Account B earns $7 in interest per month plus a one-time bonus of $55. After how many months will the total amount in both accounts be equal? Answer: 10 Solution: Let m be the number of months. Total for Account A = 9m + 35 Total for Account B = 7m + 55 Set them equal: 9m + 35 = 7m + 55 Subtract 7m from both sides: 9m - 7m + 35 = 55, so 2m + 35 = 55 Subtract 35 from both sides: 2m = 20 Divide both sides by 2: m = 10 The answer is 10 months.
   Full step-by-step solution

   Step 1: Let m be the number of months. Step 2: Total for Account A = 9m + 35 Step 3: Total for Account B = 7m + 55 Step 4: Set them equal: 9m + 35 = 7m + 55 Step 5: Subtract 7m from both sides: 9m - 7m + 35 = 55, so 2m + 35 = 55 Step 6: Subtract 35 from both sides: 2m = 20 Step 7: Divide both sides by 2: m = 10 The answer is 10 months.

4. 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.

5. Aroha is saving money to buy a new laptop that costs $980. She already has $140 saved from her birthday. She plans to save $x from her part-time job each week. After 6 weeks, she still needs $360 more to reach her goal. Write and solve an equation to find how much Aroha saves each week. Answer: 80 Solution: Let x be the amount Aroha saves each week. After 6 weeks, she saves 6x dollars. Total money after 6 weeks = initial savings + 6x = 140 + 6x.
   Full step-by-step solution

   Step 1: Let x be the amount Aroha saves each week. Step 2: After 6 weeks, she saves 6x dollars. Step 3: Total money after 6 weeks = initial savings + 6x = 140 + 6x. Step 4: She needs $980 total but still needs $360 more, so the amount she has after 6 weeks is 980 - 360 = 620. Step 5: Set up the equation: 140 + 6x = 620. Step 6: Subtract 140 from both sides: 6x = 480. Step 7: Divide both sides by 6: x = 80. The answer is 80 dollars per week.

6. Ava is saving money to buy a new bicycle that costs $210. She already has $60 saved from her birthday. She plans to save the same amount each week from her allowance. After 6 weeks, she still needs $30 more to reach her goal. How much money does Ava save each week from her allowance? Answer: 20 Solution: Let x be the amount Ava saves each week. After 6 weeks, she has saved 60 + 6x dollars. She still needs $30 more to reach $210, so her savings so far are 210 - 30 = 180.
   Full step-by-step solution

   Step 1: Let x be the amount Ava saves each week. Step 2: After 6 weeks, she has saved 60 + 6x dollars. Step 3: She still needs $30 more to reach $210, so her savings so far are 210 - 30 = 180. Step 4: Set up the equation: 60 + 6x = 180 Step 5: Subtract 60 from both sides: 6x = 120 Step 6: Divide both sides by 6: x = 20 The answer is $20 per week.

7. Noah is designing a rectangular garden on a coordinate grid. The garden has vertices at (3, 2), (3, 7), (x, 7), and (x, 2). A diagonal path is drawn from the bottom-left corner (3, 2) to the top-right corner (x, 7). The length of this diagonal path is 13 units. What is the value of x? Answer: 15 Solution: The bottom-left corner is (3, 2) and the top-right corner is (x, 7). The horizontal distance (width) is (x - 3) units. The vertical distance (height) is (7 - 2) = 5 units.
   Full step-by-step solution

   Step 1: The bottom-left corner is (3, 2) and the top-right corner is (x, 7). Step 2: The horizontal distance (width) is (x - 3) units. Step 3: The vertical distance (height) is (7 - 2) = 5 units. Step 4: The diagonal path is the hypotenuse of a right triangle, so by the Pythagorean theorem: (x - 3)^2 + 5^2 = 13^2. Step 5: Calculate 5^2 = 25 and 13^2 = 169. Step 6: The equation is (x - 3)^2 + 25 = 169. Step 7: Subtract 25 from both sides: (x - 3)^2 = 144. Step 8: Take the square root of both sides: x - 3 = 12 or x - 3 = -12. Step 9: Since x is a coordinate on the grid, distance must be positive, so x - 3 = 12. Step 10: Add 3 to both sides: x = 15. The answer is 15.

8. Liam is drawing a square on a coordinate grid. The square has vertices at (0, 0), (x + 5, 0), (x + 5, x + 5), and (0, x + 5). A diagonal is drawn from (0, 0) to (x + 5, x + 5). The slope of this diagonal is 1. What is the value of x? Answer: 0 Solution: The slope formula between two points (x1, y1) and (x2, y2) is (y2 - y1) / (x2 - x1). Here the endpoints are (0, 0) and (x + 5, x + 5). So slope = ( (x + 5) - 0 ) / ( (x + 5) - 0 ) = (x + 5) / (x + 5).
   Full step-by-step solution

   Step 1: The slope formula between two points (x1, y1) and (x2, y2) is (y2 - y1) / (x2 - x1). Step 2: Here the endpoints are (0, 0) and (x + 5, x + 5). Step 3: So slope = ( (x + 5) - 0 ) / ( (x + 5) - 0 ) = (x + 5) / (x + 5). Step 4: We are told the slope is 1, so set up the equation: (x + 5) / (x + 5) = 1. Step 5: This is true for any value of x where x + 5 is not zero. However, the square's side length must be positive, so x + 5 > 0, meaning x > -5. Step 6: But the slope simplifies to 1 for any x (except x = -5 which gives a side length of 0, not a square). The problem implies a unique solution, so we must check if there is a condition from the geometry: the diagonal of a square always has slope 1 or -1 depending on orientation. Since it's from bottom-left to top-right, slope = 1 is always true for any square with sides parallel to axes. Step 7: The problem expects a specific x. If we consider that the diagonal has slope 1 regardless of x, then any x > -5 works, but the simplest value that makes the square have non-zero side length is x = 0. Step 8: Alternatively, if we set the slope equal to 1, we get (x+5)/(x+5) = 1, which is an identity for all x except x = -5. The problem likely intends x = 0 as the simplest solution. The answer is 0.