Derive y=mx+b

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Derive y=mx+b · Grade 8 · Worksheet 3

1. Matiu is helping his family track the water level in their rainwater tank. At the start of a dry spell (day 0), the tank contained 120 liters of water. After 4 days without rain, the tank had dropped to 72 liters. Assuming the water level decreases at a constant rate, write a linear equation in the form y = mx + b that represents the amount of water y (in liters) in the tank after x days. Answer: y = -12x + 120 Solution: Identify two points from the problem: (0, 120) at the start and (4, 72) after 4 days. Calculate the slope (m) using the formula m = (y2 - y1) / (x2 - x1). m = (72 - 120) / (4 - 0) = (-48) / 4 = -12.
   Full step-by-step solution

   Step 1: Identify two points from the problem: (0, 120) at the start and (4, 72) after 4 days. Step 2: Calculate the slope (m) using the formula m = (y2 - y1) / (x2 - x1). Step 3: m = (72 - 120) / (4 - 0) = (-48) / 4 = -12. Step 4: The y-intercept (b) is the starting amount of water at day 0, which is 120. Step 5: Substitute m and b into y = mx + b: y = -12x + 120. The equation is y = -12x + 120.

2. Points (3, 17) and (9, 41). Find equation in y = mx + b form. Answer: y = 4x + 5 Solution: Find the slope m using m = (y2 - y1) / (x2 - x1). Using (3, 17) and (9, 41): m = (41 - 17) / (9 - 3) = 24 / 6 = 4. Use point (3, 17) and m = 4 in y = mx + b: 17 = 4(3) + b → 17 = 12 + b.
   Full step-by-step solution

   Step 1: Find the slope m using m = (y2 - y1) / (x2 - x1). Using (3, 17) and (9, 41): m = (41 - 17) / (9 - 3) = 24 / 6 = 4. Step 2: Use point (3, 17) and m = 4 in y = mx + b: 17 = 4(3) + b → 17 = 12 + b. Step 3: Solve for b: b = 17 - 12 = 5. Step 4: Write the equation: y = 4x + 5.

3. Find the slope of the line passing through points (4, 7) and (8, 15). Answer: 2 Solution: Identify the coordinates: (4, 7) and (8, 15) Calculate the change in y-values: 15 - 7 = 8 Calculate the change in x-values: 8 - 4 = 4 Divide the change in y by the change in x: 8 ÷ 4 = 2 The slope is 2.
   Full step-by-step solution

   Step 1: Identify the coordinates: (4, 7) and (8, 15) Step 2: Calculate the change in y-values: 15 - 7 = 8 Step 3: Calculate the change in x-values: 8 - 4 = 4 Step 4: Divide the change in y by the change in x: 8 ÷ 4 = 2 Step 5: The slope is 2.

4. Hana is filling her family's new swimming pool with a garden hose. She notices that the water depth increases at a constant rate. At 2:00 PM (2 hours after she started), the water depth is 14 inches. At 6:00 PM (6 hours after she started), the water depth is 30 inches. Write a linear equation in the form y = mx + b that represents the water depth (y) in inches after x hours of filling. Answer: y = 4x + 6 Solution: Identify the two points from the problem. The time x is the number of hours after starting, and y is the water depth. Point 1: (2, 14).
   Full step-by-step solution

   Step 1: Identify the two points from the problem. The time x is the number of hours after starting, and y is the water depth. Point 1: (2, 14). Point 2: (6, 30). Step 2: Calculate the slope m using the formula m = (y2 - y1) / (x2 - x1). m = (30 - 14) / (6 - 2) = 16 / 4 = 4. This means the water depth increases by 4 inches per hour. Step 3: Use the slope and one point to find the y-intercept b. Use y = mx + b. Substitute x = 2, y = 14, m = 4: 14 = 4(2) + b. Simplify: 14 = 8 + b. Subtract 8 from both sides: b = 6. So the starting depth at time 0 was 6 inches. Step 4: Write the equation in y = mx + b form: y = 4x + 6. The equation is y = 4x + 6.

5. Aisha is designing a wheelchair ramp for her school's new accessibility project. The ramp needs to rise 3 feet to reach the entrance. Building codes require that for every 1 foot of vertical rise, there must be at least 12 feet of horizontal run. If the ramp starts at ground level (0 feet) and follows a linear path, write the equation in the form y = mx + b that represents the height y of the ramp at any horizontal distance x from the starting point. Answer: y = (1/12)x Solution: Identify the slope (m). The building code states that for every 1 foot of vertical rise, there must be 12 feet of horizontal run. This means the slope is rise over run = 1/12.
   Full step-by-step solution

   Step 1: Identify the slope (m). The building code states that for every 1 foot of vertical rise, there must be 12 feet of horizontal run. This means the slope is rise over run = 1/12. Step 2: Identify the y-intercept (b). The problem states the ramp starts at ground level (0 feet), so when x = 0, y = 0. Therefore, b = 0. Step 3: Write the equation in slope-intercept form y = mx + b. Substituting m = 1/12 and b = 0 gives y = (1/12)x. The equation is y = (1/12)x.

6. Emma is tracking the temperature change during a science experiment. She records that at the start (time 0 minutes), the temperature is 15°C. After 8 minutes, the temperature has risen to 31°C. Assuming the temperature increases at a constant rate, write a linear equation in the form y = mx + b that represents the temperature (y) after x minutes. Answer: y = 2x + 15 Solution: Identify the two points from the problem: (0, 15) and (8, 31) Calculate the slope (m) using the formula: m = (y2 - y1) / (x2 - x1) m = (31 - 15) / (8 - 0) = 16 / 8 = 2 The y-intercept (b) is the temperature at time 0, which is 15 Substitute m and b into the equation y = mx + b y = 2x + 15 The…
   Full step-by-step solution

   Step 1: Identify the two points from the problem: (0, 15) and (8, 31) Step 2: Calculate the slope (m) using the formula: m = (y2 - y1) / (x2 - x1) Step 3: m = (31 - 15) / (8 - 0) = 16 / 8 = 2 Step 4: The y-intercept (b) is the temperature at time 0, which is 15 Step 5: Substitute m and b into the equation y = mx + b Step 6: y = 2x + 15 The equation is y = 2x + 15.

7. Liam is designing a rectangular garden. The length of the garden is 5 feet more than twice its width. If the perimeter of the garden is 82 feet, write an equation in the form y = mx + b that represents the relationship between the length (y) and width (x) of the garden. Answer: y = 2x + 5 Solution: - Width = x - Length = y The problem says: "The length of the garden is 5 feet more than twice its width." "Twice its width" means 2x. "5 feet more than twice its width" means 2x + 5.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Define variables** Let: - Width = x - Length = y --- **Step 2: Translate the first sentence into an equation** The problem says: "The length of the garden is 5 feet more than twice its width." "Twice its width" means 2x. "5 feet more than twice its width" means 2x + 5. So: y = 2x + 5 --- **Step 3: Check if we need the perimeter information** The perimeter of a rectangle is: Perimeter = 2 × (length + width) = 2(y + x) We are told the perimeter is 82 feet: 2(y + x) = 82 --- **Step 4: Substitute y from Step 2 into the perimeter equation** From Step 2: y = 2x + 5 Substitute into 2(y + x) = 82: 2( (2x + 5) + x ) = 82 --- **Step 5: Simplify and solve for x (just to verify)** 2(3x + 5) = 82 6x + 10 = 82 6x = 72 x = 12 --- **Step 6: Find y** y = 2(12) + 5 = 24 + 5 = 29 --- **Step 7: Verify with perimeter** Perimeter = 2(29 + 12) = 2(41) = 82 ✔ --- **Step 8: Conclusion** The equation relating length (y) to width (x) is already found in Step 2: y = 2x + 5 This matches the required form y = mx + b, where m = 2 and b = 5. --- **Final answer:** y = 2x + 5