How To Solve: Equation of a Line

Attached pdf of this Article as SPOILER at the top! Happy learning!

Hi All,

I have recently uploaded a video on YouTube to discuss Equation of a Line in Detail:




Following is covered in the video

¤ Equation of a Line: Two Point Form
¤ Equation of a Line: Point and Slope Form
¤ Equation of a Line: Intercept Form
¤ Generic Equation of a line (Point and Intercept Form)
¤ Equation of Horizontal and Vertical Lines

Equation of a Line: Two Point Form

Let's say we have a line passing through two point A(\(x_1\),\(y_1\)) and B(\(x_2\),\(y_2\)).
Let's take a point C (x,y) on the line and between A and B as shown below.



Slope of the Part CA of the line = Slope of the part BA of the line = Slope of the line

=> \(\frac{y - y_1}{x - x_1}\) = \(\frac{y_2 - y_1}{x_2 - x_1}\) [ Watch this video if you want to know about the slope of the line ]

=> Equation of a line in Two Point Form as y - \(y_1\) = \(\frac{y_2 - y_1}{x_2 - x_1}\) * (x - \(x_1\))

where x and y are variables and value of (\(x_1\),\(y_1\)) and (\(x_2\),\(y_2\)) will be given to us in the problem.


Equation of a Line: Point and Slope Form

Let's take the same line which was passing through point A and B and has a slope m



Slope, m = \(\frac{y_2 - y_1}{x_2 - x_1}\)

Using, y - \(y_1\) = \(\frac{y_2 - y_1}{x_2 - x_1}\) * (x - \(x_1\))

Equation of a line in Point and Slope Form as y - \(y_1\) = m * (x - \(x_1\))


Equation of a Line: Intercept Form

Let's say we have a line which intercepts X-Axis at point A(a,0) and Y-Axis at point B(0,b), as shown below.



Using, \(\frac{y - y_1}{x - x_1}\) = \(\frac{y_2 - y_1}{x_2 - x_1}\) we get

\(\frac{y - 0}{x - a}\) = \(\frac{b - 0}{0 - a}\)
=> ay = -bx + ab
=> bx + ay = ab

Dividing both the sides by ab we get

Equation of a line in Intercept Form as \(\frac{x}{a}\) + \(\frac{y}{b}\) = 1


Generic Equation of a line (Point and Intercept Form)

Let's say we have a line which intercepts Y-Axis at point B(0,b) and has a slope m, as shown below.



Using, y - \(y_1\) = m * (x - \(x_1\)) and substituting the value of Point B we get

y - b = m * (x - 0)

Generic Equation of a line (Point and Intercept Form) as y = mx + b

where m is the slope of the line and B is the y intercept.


Equation of horizontal and vertical lines

Let's say we have a line parallel to X-Axis and intersecting Y-Axis at point B(0,b) and
a line which is parallel to Y-Axis and intercepting the X-Axis at point A(a,0) as shown below



Equation of Horizontal Line

Now, all the points on this line will be at the same distance b from X-Axis and will have the y-coordinate as b

=> Equation of Horizontal line will be y = b [constant]

Equation of Vertical Line

Now, all the points on this line will be at the same distance a from Y-Axis and will have the x-coordinate as a

=> Equation of Vertical line will be x = a [constant]


Hope it helps!
Good Luck!