Write an equation of the line that passes through the given two points
A two-point form of the equation is used when two different points on the line are known.
Answer: The general equation of the line passing through the points (\(x_{1}\), \(y_{1}\)) and (\(x_{2}\), \(y_{2}\)) is y - \(y_{1}\) = [(\(y_{2}\) - \(y_{1}\)) / (\(x_{2}\) - \(x_{1}\))] (x - \(x_{1}\)).
This equation can easily be simplified to any of the forms of the equation like the slope-intercept form, so as to calculate the intercept value by comparison.
Explanation:
Let the given points are (\(x_{1}\), \(y_{1}\)) and (\(x_{2}\), \(y_{2}\))
Therefore, applying the slope-intercept form of the equation,
We get,
⇒ y - \(y_{1}\) = m (x - \(x_{1}\))
⇒ m = slope = (\(y_{2}\) - \(y_{1}\)) / (\(x_{2}\) - \(x_{1}\))
Consider an example.
Let the two given points be (1, 0) and (3, 4)
Slope of the line
= (4 - 0) / (3 - 1 )
= 4 / 2 = 2 ------------ (1)
Using the point (1, 0), let's write the equation of the line
Using the general form of the equation i.e. y - \(y_{1}\) = [(\(y_{2}\) - \(y_{1}\)) / (\(x_{2}\) - \(x_{1}\))] (x - \(x_{1}\)),
(y - 0) = m (x - 1) [Since, (\(y_{2}\) - \(y_{1}\)) / (\(x_{2}\) - \(x_{1}\)) = m]
⇒ y = 2(x - 1) [From (1), m = 2]
⇒ y = 2x - 2