Slope Intercept Form: An Introduction With Examples

Slope Intercept Form: An Introduction With Examples

In mathematics, the slope intercept form is frequently used to find the straight line equation. This equation is also known as a linear equation because the degree of each variable is one. Several methods are used to find the equation of the line.

The slope intercept form is one of them. The linear equation of straight line can be written in various forms. For example, y = 12x – 8 can be written in different ways such as:

  • 12x – y – 8 = 0
  • x = (y + 8) / 12
  • y – 12x = 8 

The equation of the line can also be determined by using the point slope form and two points intercept form. In this article, we will study all the basics of slope intercept form along with examples.

What is the slope intercept form?

The slope intercept form is a well-known technique used to determine the straight line equation by using the slope and intercept of the line. The slope and y intercept of the line must be known in order to find the line’s equation.

The equation of the slope intercept form is:

Y = m * x + b 

Where m is the slope, x & y are the fixed points, and b is the y-intercept form. 

In general, the equation of a straight line is linear then a question arises in the mind about why we use a linear equation to find the linear equation. The answer to this question is that the equation of the slope intercept form is general.

We have to substitute the points of unknown values of the equation to get the exact equation of the line. The unknown terms of the equation are slope and y-intercept form. We must be familiar with these terms.

Slope of the line

The slope of the line is the quotient of change in the values of the y coordinate and the change in the values of the x coordinate. The slope is simply the rise over run of the given Cartesian points. The equation of the slope of the line is:

Slope = m = change in the values of y (rise) / change in the values of x (run)

Slope = m = [(y2 – y1) / (x2 – x1)]

Slope = m = Δy / Δx

Y-intercept of the line

The y-intercept of the line is a point on it where the line meets the x-axis in a coordinate plane. If the term “x” in the equation is zero, then the slope intercept form is:

y = m * 0 + b

y = b

How to find the line’s equation by using the slope intercept form?

There are three methods to find the linear equation of the line by using the slope intercept form. These methods are:

  1. Two points method
  2. One point and slope method
  3. Slope and y-intercept method

Let u take some examples of the methods of the slope intercept form to find the line’s equation.

By using two points methods 

Example

Find the line’s equation if the coordinate points of the line are (x1, y1) = (12, 19) and (x2, y2) = (32, 29).

Solution

Step I: First of all, identify the given x and y values. 

x1 = 12, x2 = 32, y1 = 19, y2 = 29

Step II: Now take the general equation of slope and find the slope of the line by using the above points.

m = y2 – y1 / x2 – x1

m = 29 – 19 / 32 – 12

m = 10 / 20

m = 5/10 

m = 0.5

Step III: Take the general formula of the slope intercept form

y = m * x + b 

Step IV: Take the first pair of points and determine the y-intercept of the line.

y = m * x + b 

19 = 0.5 * 12 + b

19 = 6 + b

19 – 6 = b

13 = b

b = 13

Step V: Determine the line’s equation by substituting m = 0.5 and b = 13 in the slope intercept formula.

y = m * x + b 

y = 0.5 * x + (13)

y = 0.5 * x + 13 

y = 0.5 * (x + 6.5)

So, y = 0.5 * (x + 6.5) is the straight line equation.

The above problem of the slope intercept form can also be solved by using a slope intercept form calculator to get the solution with steps in a fraction of seconds. To learn how to calculate the line’s equation using a calculator, follow the steps below

  • Select the method i.e., two points, 1 point & slope, or slope & y-intercept.
  • Write the required values.
  • Click the calculate button. 
  • The result will come in a couple of seconds.

By using the one point and slope method

Example

Identify the linear equation of the straight line, if the slope of the equation is 2 and a point (x1, y1) = (22, 36)

Solution 

Step I: Identify the slope and given points of the line.

Slope = m = 2

Point of the line = (22, 36)

x1 = 22 

y1 = 36

Step II: Take the general formula of the slope intercept form

y = m * x + b 

Step III: Take the slope and points (x1, y1) = (22, 36) to determine the y-intercept of the line.

y = m * x + b 

36 = 2 * 22 + b

36 = 44 + b

36 – 44 = b

-8 = b

b = -8

Step IV: Determine the line’s equation by substituting m = 2 and b = -8 in the slope intercept formula.

y = m * x + b 

y = 2 * x + (-8)

y = 2 * x – 8

y = 2 * (x – 4) 

So, y = 2 * (x – 4) is the line’s equation.

By using the slope and y-intercept method

Determine the straight line’s equation if the slope m = 6 and y-intercept b = 12.

Solution 

Step I: Identify the slope ad y-intercept of the line.

Slope of the equation = m = 6

y-intercept of the equation = b = 12

Step II: Take the general formula of the slope intercept form

y = m * x + b 

Step III: Determine the straight line’s equation by substituting m = 6 and b = 12 in the slope intercept formula.

y = m * x + b 

y = 6 * x + 12

y = 6 * (x + 2) 

So, y = 6 * (x + 2) is the straight line equation. 

Summary 

In this article, we have discussed all the basics of the slope intercept form with methods and examples. Now after reading the above post, you can solve any problem related to this topic easily. 

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