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

Thus, the equation of the line passing through the points (1, 0) and (3, 4) is y = 2x - 2.