Finding the Equation of a Line
To find the equation of a line y=ax+b, both the slope a and the intercept b are needed. Here are methods for finding them from given conditions.
When the Slope and One Point Are Given
Example ①
Find the equation of the line with slope 3 passing through the point (2, 5).
Answer: Substitute (2,5) into y=3x+b: 5=6+b → b=−1. y=3x−1
Find the equation of the line with slope 3 passing through the point (2, 5).
Answer: Substitute (2,5) into y=3x+b: 5=6+b → b=−1. y=3x−1
When Two Points Are Given
Example ②
Find the equation of the line passing through the points (1, 2) and (3, 8).
Answer: Slope a=(8−2)/(3−1)=3. Substitute (1,2) into y=3x+b: 2=3+b → b=−1. y=3x−1
Find the equation of the line passing through the points (1, 2) and (3, 8).
Answer: Slope a=(8−2)/(3−1)=3. Substitute (1,2) into y=3x+b: 2=3+b → b=−1. y=3x−1
Key Points
- Slope a = (y₂−y₁)/(x₂−x₁) (difference in y-values ÷ difference in x-values)
- Once a is known, substitute one point into y=ax+b to find b
Practice Problems
- Find the equation of the line with slope −2 passing through the point (1, 3).
- Find the equation of the line passing through the points (−1, 5) and (3, −3).
Answers and Explanations
- Answer: y=−2x+5. 3=−2+b→b=5.
- Answer: y=−2x+3. a=(−3−5)/(3−(−1))=−8/4=−2. Substitute (3,−3): −3=−6+b→b=3.