What is slope intercept form and what are its applications?

In algebra, a slope intercept form is an equation that is written in the form of y = MX + b. In this equation, we have two variables (x and y) and two constants (m and b). Slope intercept form follows the properties of a linear function in which an algebraic equation consists of a variable (whose power is one) and a fixed number or a product of fixed numbers (constants).

The slope intercept form is also called a linear function. For a better understanding of linear function, we are elaborating this as under.

What is Linear Function?

A linear function is a function whose graph is always a straight line and has a combination of one dependent and independent variable. The variable on the left side of the equation is called the independent variable whereas the variable on the right side of the equation is known as the dependent variable. Just like polynomial function, the degree of variable is 0 or 1 in a linear function. The formula or expression of the linear function is the same as of slope intercept form e.g.,

y = b + mx.

Graph of a Linear Function

The graph of the above linear equation is a straight line on the plane where b and m are constants.

The graphs of the following equations will also be a straight line.

y = 5x + 7

y = -2x – 5

Slope-Intercept of Straight line or Linear Function

In the linear function, the fixed number m is called the slope or gradient of line and b denotes the y-intercept (a point where the line crosses the y-axis). It is also called the vertical intercept.

There are many possibilities of the slope of the line that is shown on the graph which is described below:

1. If the value of m is positive, the slope of the line will be in an upward direction starting from the left bottom and ending at the right upper side of the plane.
2. If the value of m is negative, the slope of the line will be in a downward direction from the left upper side to the right bottom side of the plane.
3. But if the value of m is zero, the line will be horizontal.
4. When x=0, then the line will draw on a vertical axis.

Graphical representation of slope intercept

If the line has a negative gradient, the graph will be as below:

If the line has a positive gradient, the graph will draw as:

If the line has zero gradients, the graph will be shown as:

 

Applications of slope intercept:

Example 1: Find the vertical intercept and gradient of the line?

y = 16x – 8

Solution:

Compare the given equation with linear equation y = mx – b

Where

Gradient = m= 16

y-intercept = b = -8

Points (0, -8)

Here, the value of m is positive, the slope of the line will be in an upward direction starting from the left bottom and ending at the right upper side of the plane.

Example 2: If the slope of the line is -7 and the y-intercept is -5, then write the equation in slope-intercept form?

Solution:

Given as

Slope = m= -7

y-intercept = b= -5

Put the values in the equation

y = mx + b

y = (-7)x + (-5)

y = -7x – 5

Points derived (0,-5)

Here, the value of m is negative, the slope of the line will be in a downward direction from the left upper side to the right bottom side of the plane.

Example 3: How can we write the line equation for points (0, -3) if the slope-intercept is passing through with a slope of 7?

Solution:

Identifying the values

m = 7

y = -3

x = 0

First of all, we will find the missing value of y-intercept (b) by putting the known values to the equation of slope-intercept form

y = mx + b

-3 = 7(0) + b

-3 = 0 + b

-3 = b

The line equation or slope intercept form can be written as

y = mx + b

y = (7)x + (-3)

y = 7x – 3

Example 4: Write the line equation for the slope of passing through the point (-3, -3)?

Solution:

The given values are

m =

From the points (-3, -3) we can take the values of x & y

x = -3

y = -3

To find the value of b, put the known values in slope-intercept form

y = mx + b

-3 = (-3) + b

-3 = + b

-3 – = – + b

-3 – = b

– = b

= b

= b

Now all values are given just put the values in slope intercept form

y = mx + b

y = ()x +

y = –

Slope Formula

The slope of a line can be found by diving the coordinates at the y-axis with the coordinates of the x-axis by taking the difference between ending & starting points.

In equation form

m =

Example 1: Find the slope, y-intercept and write the line equation for the points (6,7) and (0,2)?

Solution:

The slope formula is as under:

m =

From the given points we can derive the value of x₁, x₂, y₁ & y₂

1^st Point is (6, 7)

In which x₁ = 6 and y₁ = 7

2^nd Point is (0, 2)

Here x₂ = 0 and y₂ = 2

To simplify the formula, put the known values in the slope formula

m =

m =

Find the y-intercept for first points (6, 7)

m = =

x₁ = 6 and y₁ = 7

y = mx + b

7 =

7 =

7 – 5 = b

2 = b

Find the y-intercept for Second points (0, 2)

m =

x₂ = 0 and y₂ = 2

y = mx + b

2 =

2

2 = b

Now write the linear equation in slope intercept form

y = MX + b

y =

Graphical representation of the slope intercept form:

x

y

y =

(6,7)

(0,2)

The above calculation steps are used to determine the required values of slope intercept form of linear equation. Whereas, many online algebraic tools are also available to solve the linear equation and to find the values of the slope of the line, y-intercept, and x-intercept such as slope intercept form calculator. By using this online tool, you can simply put the known values in the text boxes and find out the required information instantly.