How to Derive the Slope Intercept Form?

We can derive the slope intercept form of any line:

• From Point-Slope Form
• From Two Point Form

Let’s discuss these methods in detail.

Derivation from Point-Slope Form

The slope intercept of the line is easily derived using the formula for the equation of the straight line passing through a fixed coordinate and making the slope m with the x-axis.

Now to prove this take a line L with slope m with the x-axis and which cuts the y-axis at a distance of c units from the origin.

The distance ‘c’ on the y-axis is the y-intercept and the coordinate of the line where it cuts the y-axis is (0, c). It is given that the slope of the line is m.

We know that the equation of a line in point-slope form, that is passing through a point (x₁, y₁) and having the slope m is,

(y – y₁) = m(x – x₁)…(i)

Given, (x₁, y₁) = (0, c)

Putting this value in eq(i), we get;

y – c = m(x – 0)

y – c = mx

y = mx + c

This is the equation of the straight line in Intercept form. Here, m represents the slope of the straight line, and c represent the intercept made by the line on the y-axis.

Derivation from Two Point Form

The Two-Point Form of a linear equation is given by:

[Tex]y – y_1 = \frac{y_2 – y_1}{x_2 – x_1} (x – x_1)[/Tex]

⇒ [Tex]y = \left(\frac{y_2 – y_1}{x_2 – x_1}\right) x – \left(\frac{y_2 – y_1}{x_2 – x_1}\right)x_1 + y_1[/Tex]

As [Tex]- \left(\frac{y_2 – y_1}{x_2 – x_1}\right)x_1 + y_1[/Tex] is constant, thus let [Tex]C = – \left(\frac{y_2 – y_1}{x_2 – x_1}\right)x_1 + y_1[/Tex]

Therefore, [Tex]y = \left(\frac{y_2 – y_1}{x_2 – x_1}\right) x + C[/Tex]

As we know that slope of any line passing through two points is given as: [Tex]m = \frac{y_2 – y_1}{x_2 – x_1}[/Tex]

Thus, y = mx + C

Slope Intercept Form of a line is one of the many standard forms of equation of line. Equation of line is the algebraic way to represent a line.The various ways of representing the equation of a straight line are,

• Slope intercept form
• Intercept form
• Point slope form
• Two point form

Slope intercept form of the equation is used to represent the equation of a straight line when the slope of the equation and the intercept on the y-axis is given.

In this article, we will learn about the slope-intercept form of a straight line, its derivation, graphs, and examples.