Slope Intercept Form – Definition, Formula, Facts, Examples

What Is the Slope-Intercept Form of a Straight Line?

The slope-intercept form of the equation of a straight line is used to write the equation of a line using its slope and the y-intercept. It is usually given by y = mx + b.

The slope of a line is given by the rise-over-run ratio. The y-intercept is the point where the line intersects with the Y-axis.

Slope-Intercept Form: Definition

The slope intercept form is a way of writing the equation of the straight line using the slope and the y-intercept of the line.

The equation of a line with slope m and y-intercept b is written in the slope-intercept form as

$y = mx + b$ 

where

(x,y) represents the coordinates of any point on the line.

However, the slope-intercept formula of the straight line cannot be used to write the equation of a vertical line because the slope of the vertical line is not defined.

Slope-Intercept Form: Formula

The equation of a straight line in the slope-intercept form is given by 

$y = mx + b$

Where

m equals the slope

b represents the y-intercept of the straight line 

(x,y) determines each point on the straight line and is considered as variable.

Derivation of Formula For Slope-Intercept  Form

Let us assume that a line with a slope m has the y-intercept b. It means that the line intersects the y-axis at the point (0, b).

Let (x,y) be another random point on the line. 

Thus, we have the coordinates of two points on the line.

$(x_{1},\; y_{1}) = (0,\;b)$ and $(x_{2},\; y_{2}) = (x,\;y)$

Alt Text: Graphical representation of the slope-intercept form equation of a straight line.

We know that the slope of a line passing through the points $(x_{1},\; y_{1})$ and $(x_{2},\; y_{2})$ is given by

$m = \frac{(y_{2}\;-\;y_{1})}{(x_{2}\;-\;x_{1})}$

For $(x_{1},\; y_{1}) = (0,\;b)$ and $(x_{2},\; y_{2}) = (x,\;y)$, we can write

$m = \frac{(y \;-\; b)}{(x \;-\; 0)}$

$m = \frac{(y \;-\; b)}{x}$

$mx = (y\;-\;b)$

$y = mx + b$

This is called the slope-intercept form of the equation of a straight line.

Slope-Intercept Form: Examples

• Suppose the slope is $-2$ and the y-intercept is 5. The equation of the line is given by

$y = \;-\;2x + 5$

• Slope $= 5$ and the line passes through the origin (0, 0). The equation of line is given by

$y = 5x$

Straight Line Equation Using Slope-Intercept Form

To evaluate the line equation with an arbitrary inclination, two quantities are required, i.e., 

• Inclination/slope of the line
• Arrangement of the line based on the coordinates of each point on the y-axis

Lines can be formed using these two key parameters. Let us know the steps to evaluate what the slope-intercept form of a line is.

Step 1: Find the slope (m) of the line using the given information.

• If θ is the angle the line makes with the positive x-axis, the slope of the straight line = tan θ. 
• If $(x_{1},\; y_{1})$ and $(x_{2},\;y_{2})$ are two points on the line, then the slope of the straight line $= \frac{(y_{2}\;-\;y_{1})}{(x_{2}\;-\;x_{1})}$.

Step 2: Note down the y-intercept (b). 

Step 3: Substitute the values in the slope-intercept form $y = mx + b$ to find the equation of the straight line.

Converting Standard Form to slope-intercept Form

A standard form equation of a line can easily be converted to the slope-intercept form by comparison and rearrangement of the points. Let us explore the standard equation represented as follows:

$Ax + By + C = 0$ 

where A, B, C are constants; 

A, B cannot be simultaneously 0.

Let’s rewrite it as

$By = \;-\; Ax\;-\;C$

B rearranged from LHS to RHS, i.e., from multiplication to division on the other side, we get

$y = (\frac{-\;A}{B})x + (\frac{-\;C}{B})$

Therefore, we get

• slope $= m = (\frac{-\;A}{B})$ 
• y-intercept $= b = (\frac{-\;C}{B})$

Facts about the Slope-Intercept Form

Conclusion

In this article, we learned about the slope-intercept form, which is used to find the equation of the straight line using the slope and the y-intercept. It is given by y = mx + b. We learned the formula, and its derivation. Let’s solve a few examples and practice problems to master these concepts.

Solved Examples of Slope-Intercept  Form of Line

1. Evaluate the straight line equation where slope $m = 4$ passes via the point $(\;-\;1, \;-\;3)$.

Solution: 

Let the equation of line be $y = mx + c$.

Slope of the line is $m = 4$

The line passes through the point $(\;-\;1,\; -\;3)$. Let’s use it to find the y-intercept.

Substitute $y = \;-\;3$ and $x = \;-\;1$ in $y = mx + c$.

Putting the values in the above slope-intercept formula, we will obtain

$\;-\;3 = 4(\;-\;1) + b$

$\;-\;3 = \;-\;4 + b$

$b = \;-\;3 + 4$

$b = 1$

Thus, we have $m = 4$ and $b = 1$.

Using the slope-intercept form, we write the equation of the line as

$y = 4x + 1$

2. Evaluate the equation of the straight line when $m = \;-\;2$ and passes through the point $(3,\; -\;4)$.

Solution: 

Let the equation of line be $y = mx + b$.

The line passes through $(3,\; -\;4)$.

Thus, the point satisfies the equation.

$-\;4 = \;-\;2 (3) + b$

$\;-\;4 = \;-\;6 + b$

$b = \;-\;4 + 6$

$b = 2$

Thus, the y-intercept is 2.

Therefore, the required slope-intercept form equation of the straight line will be written as $y = \;-\;2x + 2$

3. Write the equation of line $7x + 8y \;-\; 1 = 0$ in the slope-intercept form. Find the slope and y-intercept.

Solution: 

We want to write the equation of the line in the form $y = mx + b$.

$7x + 8y \;-\; 1 = 0$

$\Rightarrow 8y = \;-\;7x + 1$

$\Rightarrow y = \frac{-\;7x}{8} + \frac{1}{8}$

The slope of the line is $\frac{-\;7x}{8}$. 

The y-intercept is $\frac{1}{8}$.

Practice Problems on slope-intercept Form of a Line

Frequently Asked Questions on Slope-intercept Form of a Line