# 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) = (y**_{2}** – y**_{1}**) / (x**_{2}** – x**_{1}**)**

**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 **(x**_{1}**, y**_{1}**) = (0, b) **and** (x**_{2}**, y**_{2}**) = (x, y)**

In addition, we know that if a line passes through the points **(x**_{1}**, y**_{1}**)** and **(x**_{2}**, y**_{2}**)** then the slope of a line can be described as:

**m = (y**_{2}** – y**_{1}**) / (x**_{2}** – x**_{1}**)**

After multiplying **(x**_{2}** – x**_{1}**) **with both sides, we get

m (x_{2} – x_{1}) = (y_{2} – y_{1})

**∴**** ****(x**_{1}**, y**_{1}**) = (0, b) and (x**_{2}**, y**_{2}**) = (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.