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Understanding Slope-Intercept Form:  Explanation, Derivation, and Examples

Tuesday - November 28, 2023

In mathematics, the slope-intercept form is a way of writing an equation for a line in the form y = mx + k. The m denotes the slope of the line and the k denotes the y-intercept. The slope intercept form is widely used when you want to calculate either a point on a line or calculate for y if you know x.

Now, we calculate the slope-intercept form, you have to calculate the slope of the line and the y-intercept form of the equation. The measure of the steepness of the line is called slope. A slope is mostly used in algebra. We calculate the slope by using the general equation of the slope.

 A slope can be calculated by using a fraction of the change in the points of y (rise) over the change in the points of x (run). In conclusion, write the equation, substituting numerical values in the form m and k. 

In this article, we will discuss the definition and derivation of slope intercept form with examples.

What is slope-intercept form?

A slope-intercept form of a straight line is used to calculate the equation of a line. For the slope-intercept formula, we have well-known the slope of the line and the intercept divided by the line with the y-axis. Consider a straight line of slope ‘m’ and y-intercept ‘k’. The slope intercept form equation for a straight line with a slope, ‘m’, and ‘k’ as the y-intercept can be given as 

An equation of a line can be written in many different methods and each of these ways is satisfied. A slope denoted how fast the line changes from point to point, while k denoted the starting point of the line. A straight line’s slope-intercept form is amongst the most common ways to describe its equation.

At this point on the y-axis where the line with a certain slope crosses or intersects is the y-intercept. Now, we write an equation in slope-intercept form, given a graph of that equation, pick two points on the line, and use them to find the slope.

Next, we find the coordinates of the y-intercept–this should be of the form (0, k). To y- coordinate is the value of k in the equation.

Equation of Slope Intercept Form  

A slope-intercept equation can be written as,

Y = mx + k

  • Where x and y are the coordinate points
  • m is the slope of the line, and  
  • k is the y-intercept.

Derivation of Slope-Intercept Form Equation

To calculate the slope-intercept form of the line equation from the equation of a straight line in the standard form as given below, as we know that, the standard form of the equation of a straight line is,

Px + Qy + R = 0

Simplify the equation. 

Qy = -Px – R

y = (-P/Q)x + (-R/Q)

The slope intercept form can be written as

 y = mx + k

Here, (-P/Q) denotes the slope of the line and (-R/Q) is the y-intercept.

Example Section 

In this section, we explain some example

Example 1:

Show the equation of the straight line of (7, 15) and (5, 8) by using a slope-intercept form.

Solution:

Step 1:

First of all, identify the given points.

x1 = 7, x2 = 5, y1 = 15, y2 = 8

Step 2:

Calculate the slope of the line using the given points. We know that, the slope of the line,

m =  ∆y/∆x

m = (y2 – y1 )/( x2 – x1 )

m = (8 – 15)/(5 – 7)

m = (- 7)/(-2)

m = 7/2

m = 3.50

Step 3:

we write the general form of slope-intercept form.

y = mx + k

Putting the calculated slope in the formula of slope-intercept form.

y = mx + k

y = 3.50x + k

Step 4:

Calculate the y-intercept (k). Let’s choose the first point, (7, 15) for calculating the y-intercept.

y = 3.50x + k

15 = 3.50(7) + k

15 = 24.50 + k

k = 15 – 24.50 

k = -9.50

y-intercept (k)= -9.50

Step 5: 

Substitute the calculated values in the formula of the slope-intercept form.

y = mx + k

y = 3.50x – 9.50

You can also use a slope and y intercept calculator to find the straight-line equation according to the equation of slope intercept form with steps in a fraction of seconds.

Example 2:

Write the following slope intercept form equation of a line in standard form:

y = x/2 – 3

Solution:

Step 1:

First of all, we write the given equation.

y = x/2 – 3 

Step 2:

Multiply each side by (2)

2y = x – 6 

Step 3:

Now, Subtract x from each side.

-x + 2y = -6 

Step 4:

Again, multiply each side by  (-1).

x – 2y = 6

Hence, the standard form equation.

Conclusion

In this article, we have discussed the basic definition of slope intercept form, the equation of slope intercept form, the Derivation formula of slope intercept form, and also with the help of an example topic will be explained. After studying this article everyone defends this topic. 

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