Slope-Intercept Form: Formula, Derivation, & Examples

The slope-intercept form is the most common way to represent a straight-line equation. This form is very helpful for determining the equation of a line when the slope of a straight line and the y-intercept point (The point at the y-coordinate where the straight line cuts the y-axis) are known.

A straight-line equation is an equation that is satisfied by each point of the line. In algebra, the equation of a line can be expressed in many different forms and each of them is useful in its own way. Some useful forms of the equation of a line are:

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

In this article, we will confine ourselves to only slope-intercept form. We will learn about slope-intercept form, its derivation, and methods to solve problems.

Definition of Slope-Intercept Form with Explanation

The slope intercept form is used to determine the equation of a straight line. To find the equation of a line through slope intercept form, you have to know the slope and the point that the intercept at the y-axis. Many students give preference to this form due to its simplicity.

It is important to note that the slope-intercept form is only applicable to linear equations. To understand the slope-intercept form better, let us divide it into two concepts.

Slope

A slope is the ratio of the vertical change (y-axis) to the horizontal change (x-axis) between two points on a straight line. Symbolically, it is expressed by “m”. Tan θ is equal to the slope of the straight line. The mathematical representation of slope is:

m = tan θ = Rate change in y-axis / Rate change in x-axis

Slope (m) = (y₂ – y₁) / (x₂ – x₁)

Y-intercept

The y-intercept is the point where any line intersects the y-axis in a coordinate plane. Its coordinates are represented as (0, b). That is the value of y when x will be equal to zero. In other terms, a y-intercept is a point where the line crosses the y-axis at the point where x = 0 and y = b.

Formula of Slope-Intercept Form

The following formula can be used to determine the equation of a straight line.

y = mx + b

Where,

• x and y are the coordinates of any point on the line.
• m is the slope (or steepness) of a straight line.
• b represents the y-intercept, a point where a line intersects the y-axis when x=0.

How to derive the formula of the Slope-Intercept form?

Let’s derive the equation of slope-intercept form using the concept of the slope.

Consider a line L with a slope m intersecting the y-axis at a length of b units from the origin O.

Suppose that (x₁, y₁) = (0, b) and (x₂, y₂) = (x, y)

In addition, we know that if a line passes through the points (x₁, y₁) and (x₂, y₂) then the slope of a line can be described as:

m = (y₂ – y₁) / (x₂ – x₁)

After multiplying (x₂ – x₁) with both sides, we get

m (x₂ – x₁) = (y₂ – y₁)

∴ (x₁, y₁) = (0, b) and (x₂, y₂) = (x, y)

m (x – 0) =  (y – b)

mx = y – b

Add “b” on both sides.

mx + b = y – b + b

mx + b = y

or

y = mx + b

That is the formula of slope-intercept form.

Steps to Find the equation of a Straight line using slope-intercept Form

Sept 1: Find the slope “m” and y-intercept (b) of the line.

Sept 2: Substitute the value of m and b into the formula of slope intercept form (i.e. y = mx + b)

Sept 3: Simplify the obtained equation until it converts into a general form.

Slope-Intercept Form Examples

Here are some examples of finding the equation of a line using the slope-intercept formula.

Example 1:

Find the equation of a line with a slope of 2 and a y-intercept of 3.

Solution:

Sept 1: Identify the slope (m) of the line and y-intercept (b).

Given:

Y-intercept = b = 3

Slope = m = 5

Sept 2: Put the values of m and b in the formula of the slope-intercept form.

∴ y = mx + b

y = 5x + 3

Therefore, that is a required equation of a straight line.

Example 2:

Determine the equation of a straight line that passes through the point (2, 1) with slope – 1.

Solution:

Sept 1: Identify the slope (m) of the line and y-intercept (b).

Given,

Slope = m = – 1

x = 2

y = 1

b is not given here, so we have to find the y-intercept.

Sept 2: To find b, put the above values in the slope-intercept equation.

1 = (-1) (2) + b

1 = – 2 + b

3 = b

Sept 3: Substitute the calculated value of b and slope into the formula of slop-intercept form.

y = – 1x + 3

That is the required straight-line equation, which passes through the point (2, 1) with slope – 1.

The problems of finding straight-line equations through slope intercept form can also be solved with the help of a slope and y intercept calculator.

Conclusion

In this article, we have learned about the slope-intercept form, which is the one of famous methods among the four other different ways to find the equation of a straight line. We examined the formula of the slope-intercept form and learned how to derive it.

We explored easy steps to find the equation of the straight line through the slope-intercept form. Furthermore, we have included multiple examples to facilitate a practical understanding of determining linear equations.