Slope Intercept Form – Definition, Formula, Facts, Examples

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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 $(x_{2},\; y_{2}) = (x,\;y)$

Slope Intercept Form – Definition, Formula, Facts, Examples

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

  • The slope-intercept form is also written as $y = mx + c$, where m is the slope and c is the y-intercept.
  • The slope-intercept form of the equation of a line having slope m and passing through the origin is $y = mx$.

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

Slope Intercept Form - Definition, Formula, Facts, Examples

Attend this quiz & Test your knowledge.

1

What is the slope-intercept form of a line formula?

$y = m + xb$
$y = x + mb$
$y = mx + b$
$y = mx \times b$
CorrectIncorrect
Correct answer is: $y = x + mb$
The slope-intercept form of a line is given by $y = mx + b$.
2

If slope $= 1$ and y-intercept $= 1$, the equation of line is

$y = 1$
$x + y = 1$
$y = x + 1$
$y = x \;-\; 1$
CorrectIncorrect
Correct answer is: $y = x + 1$
Substitute $m = 1$ and $b = 1$ in $y = mx + b$, we get $y = x + 1$.
3

The equation of a line with slope m and passing through the origin is

$y = m + x$
$y = mx$
$y = m$
$my = x$
CorrectIncorrect
Correct answer is: $y = mx$
The slope of a line with slope m that passes through the origin is $y = mx$.
4

What will be the y-intercept of equation $2x + 5y \;-\; 1 = 0$?

$\frac{1}{5}$
$\frac{2}{5}$
$-\;\frac{2}{5}$
$-\;\frac{1}{5}$
CorrectIncorrect
Correct answer is: $\frac{1}{5}$
$5y = 1\;-\; 2x$
$y = \frac{\;-\;2x}{5} + \frac{1}{5}$
Thus, y-intercept $= \frac{1}{5}$

Frequently Asked Questions on Slope-intercept Form of a Line

We use the slope-intercept form to find the equation of the line when the slope and the y-intercept is known or can be calculated using the available information.

The slope of the line determines if the line is increasing or decreasing and how steep it is established. It is denoted by m and represents how quickly y-axis coordinates change with the slight change in x-axis coordinates.

Slope-intercept form and point-slope form are two different forms to write the equation of a straight line. 

Point-slope form: $y \;-\; y_{1} = m(x \;-\; x_{1})$, where m the slope and $(x_{1},\;y_{1})$ are the coordinates of any arbitrary point on the line.

slope-intercept form: $y = mx + b$, where m the slope and b is the y-intercept.

The slope-intercept form is not used for vertical lines, as the slope of a vertical line is not defined.

We can easily identify the slope and the y-intercept of the line by looking at the slope-intercept form.