Derive y=mx+b

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Derive y=mx+b · Grade 8 · Worksheet 1

1. Find the slope of the line that passes through points (3, 7) and (8, 22). Answer: 3 Solution: Identify the coordinates: (x₁, y₁) = (3, 7) and (x₂, y₂) = (8, 22) Apply the slope formula: m = (y₂ - y₁)/(x₂ - x₁) Substitute the values: m = (22 - 7)/(8 - 3) Calculate the numerator: 22 - 7 = 15 Calculate the denominator: 8 - 3 = 5 Divide: 15 ÷ 5 = 3 The slope is 3.
   Full step-by-step solution

   Step 1: Identify the coordinates: (x₁, y₁) = (3, 7) and (x₂, y₂) = (8, 22) Step 2: Apply the slope formula: m = (y₂ - y₁)/(x₂ - x₁) Step 3: Substitute the values: m = (22 - 7)/(8 - 3) Step 4: Calculate the numerator: 22 - 7 = 15 Step 5: Calculate the denominator: 8 - 3 = 5 Step 6: Divide: 15 ÷ 5 = 3 Step 7: The slope is 3.

2. Find the slope (m) and y-intercept (b) of the line that passes through points (2, 5) and (4, 9), then write the equation in y = mx + b form. Answer: y = 2x + 1 Solution: Calculate the slope using the formula m = (y₂ - y₁)/(x₂ - x₁) Using points (2, 5) and (4, 9): m = (9 - 5)/(4 - 2) = 4/2 = 2 Use the slope and one point to find the y-intercept Using point (2, 5) and m = 2 in y = mx + b: 5 = 2(2) + b 5 = 4 + b b = 5 - 4 = 1 Write the equation with m = 2 and b = 1…
   Full step-by-step solution

   Step 1: Calculate the slope using the formula m = (y₂ - y₁)/(x₂ - x₁) Using points (2, 5) and (4, 9): m = (9 - 5)/(4 - 2) = 4/2 = 2 Step 2: Use the slope and one point to find the y-intercept Using point (2, 5) and m = 2 in y = mx + b: 5 = 2(2) + b 5 = 4 + b b = 5 - 4 = 1 Step 3: Write the equation with m = 2 and b = 1 y = 2x + 1 The equation of the line is y = 2x + 1.

3. Points (3, 13) and (7, 29) lie on a line. Derive the equation in y = mx + b form. Answer: y = 4x + 1 Solution: Find the slope m using the formula m = (y2 - y1) / (x2 - x1). Using points (3, 13) and (7, 29): m = (29 - 13) / (7 - 3) = 16 / 4 = 4. Use point (3, 13) and m = 4 in y = mx + b: 13 = 4(3) + b → 13 = 12 + b.
   Full step-by-step solution

   Step 1: Find the slope m using the formula m = (y2 - y1) / (x2 - x1). Using points (3, 13) and (7, 29): m = (29 - 13) / (7 - 3) = 16 / 4 = 4. Step 2: Use point (3, 13) and m = 4 in y = mx + b: 13 = 4(3) + b → 13 = 12 + b. Step 3: Solve for b: b = 13 - 12 = 1. Step 4: Write the equation: y = 4x + 1. The answer is y = 4x + 1.

4. A trapezoid is drawn on a coordinate plane with vertices at (1, 2), (5, 2), (7, 6), and (1, 6). A line is drawn connecting the midpoints of the two non-parallel sides. What is the equation of this line in slope-intercept form (y = mx + b)? Answer: y = 1x + 4 Solution: Identify the non-parallel sides. The trapezoid has vertices (1,2), (5,2), (7,6), (1,6). The non-parallel sides are from (1,2) to (1,6) and from (5,2) to (7,6).
   Full step-by-step solution

   Step 1: Identify the non-parallel sides. The trapezoid has vertices (1,2), (5,2), (7,6), (1,6). The non-parallel sides are from (1,2) to (1,6) and from (5,2) to (7,6). Step 2: Find the midpoint of the left side from (1,2) to (1,6). Midpoint = ((1+1)/2, (2+6)/2) = (1, 4). Step 3: Find the midpoint of the right side from (5,2) to (7,6). Midpoint = ((5+7)/2, (2+6)/2) = (6, 4). Step 4: Calculate the slope between these midpoints (1,4) and (6,4). Slope = (4-4)/(6-1) = 0/5 = 0. Step 5: Use point-slope form with slope 0 and point (1,4): y - 4 = 0(x - 1). Step 6: Simplify: y - 4 = 0, so y = 4. Step 7: Write in slope-intercept form: y = 0x + 4. The equation of the line is y = 4.

5. Emma is tracking the temperature change during a science experiment. She records that at the start (time 0 minutes), the temperature is 12°C. After 8 minutes, the temperature has risen to 28°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 + 12 Solution: Identify the two points from the problem. At time 0 minutes, temperature is 12°C, giving point (0, 12). At time 8 minutes, temperature is 28°C, giving point (8, 28).
   Full step-by-step solution

   Step 1: Identify the two points from the problem. At time 0 minutes, temperature is 12°C, giving point (0, 12). At time 8 minutes, temperature is 28°C, giving point (8, 28). Step 2: Calculate the slope (m) using the formula m = (y2 - y1) / (x2 - x1). m = (28 - 12) / (8 - 0) = 16 / 8 = 2. Step 3: The y-intercept (b) is the temperature at time 0, which is 12. Step 4: Substitute m and b into the equation y = mx + b. y = 2x + 12. The equation is y = 2x + 12.

6. Emma is training for a marathon and tracks her running distance. In her first week, she ran a total of 15 miles. Each subsequent week, she increases her distance by 2 miles. Write a linear equation in the form y = mx + b that represents the total distance (y) Emma runs after x weeks. Answer: y = 2x + 15 Solution: Identify the initial value (b). In the first week (x=0), Emma ran 15 miles. So, b = 15.
   Full step-by-step solution

   Step 1: Identify the initial value (b). In the first week (x=0), Emma ran 15 miles. So, b = 15. Step 2: Identify the rate of change (m). Each week, she increases her distance by 2 miles. So, m = 2. Step 3: Write the equation in the form y = mx + b. Substitute m = 2 and b = 15 into the equation: y = 2x + 15. The final equation is y = 2x + 15.

7. Emma is tracking the temperature change during a science experiment. She records that at the start (time 0 minutes), the temperature is 12°C. After 8 minutes, the temperature has risen to 36°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 = 3x + 12 Solution: In a linear equation of the form y = mx + b, the slope (m) represents the rate of change, and the y-intercept (b) represents the starting value when x is zero.
   Full step-by-step solution

   In a linear equation of the form y = mx + b, the slope (m) represents the rate of change, and the y-intercept (b) represents the starting value when x is zero. You can find the slope by calculating how much y changes for each unit change in x.

8. A triangle is drawn on a coordinate plane with vertices at (1, 2), (5, 6), and (9, 2). Find the equation of the line that passes through the midpoint of the base and the opposite vertex in slope-intercept form (y = mx + b). Answer: y = 2 Solution: In coordinate geometry, the base of a triangle is often the longest side or the side that appears horizontal or vertical. The midpoint formula helps find the center point of a segment.
   Full step-by-step solution

   In coordinate geometry, the base of a triangle is often the longest side or the side that appears horizontal or vertical. The midpoint formula helps find the center point of a segment. When two points share the same y-coordinate, the line between them is horizontal, which has a special slope value. Horizontal lines have a consistent equation form where y equals a constant.