Slope-intercept form: Definition, equation, and its derivation 

The slope-intercept form is a linear equation in the form of y = mx + b, where ‘m’ represents the slope of the line and ‘b’ represents the y-intercept. The slope-intercept form is an important of equation which is commonly used of a linear equation, and it provides a simple way to graph a linear equation and interpret the equation’s meaning.

The slope ‘m’ represents the rate of change of the line, or how much the line rises or falls for every one unit of horizontal distance. 

The slope-intercept form allows us to quickly and easily graph a linear equation by plotting the y-intercept first and then using the slope to determine other points on the line. It is also useful for finding the equation of a line given its slope and y-intercept.

In this article, we will discuss the definition and how to derive slope intercept form also with the help of example topic will be explained.

What is slope intercept form?

The slope-intercept form of a linear equation is a way of representing a linear relationship between two variables, usually written as y = mx + b. In this form, ‘y’ represents the dependent variable, ‘x’ represents the independent variable, ‘m’ represents the slope of the line, and ‘b’ represents the y-intercept.

The slope of a line is a measure of its steepness and can be calculated as the ratio of the change in the y-value (rise) to the change in the x-value (run) between any two points on the line. In the slope-intercept form, ‘m’ is the coefficient of ‘x’ and represents the rate of change of the dependent variable ‘y’ with respect to the independent variable ‘x’.

The y-intercept ‘b’ is the point on the y-axis where the line intersects, and is the value of ‘y’ when ‘x’ is equal to zero. It represents the starting point of the line, or the value of ‘y’ when ‘x’ is at its initial value.

Equation of Slope Intercept Form  

General equation of slope intercept form can be written as

Y = mx + k

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

Standard 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 general form of slope intercept form can also have written as

y = mx + k

Here, (-P/Q) denoted 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, we write the given equation.

x₁ = 7, x₂ = 5, y₁ = 15, y₂ = 8

Step 2:

Solves the slope of a line at a given point. 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

y=mx+b calculator can also be used to find the equation of a line with the help of coordinate points of x & y to avoid manual calculations.

Example 2:

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

y = x/2 – 3

Solution:

Step 1:

First 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.

Example 3:

Let’s say we want to write the equation of a line that has a slope of 2 and passes through the point (3, 5).

Solution :

Step 1:

Write the slope-intercept form of the equation

y = mx + b

Step 2:

Plug in the values for the slope (m) and the coordinates of the point (x, y)

 y = 2x + b

Step 3:

Solve for the y-intercept (b) by plugging in the x and y values of the given point

 5 = 2(3) + b

Step 4:

Simplify and solve for b

 5 = 6 + b

b = -1

Step 5:

Write the final equation in slope-intercept form by plugging in the values for m and b

y = 2x – 1

Step 6:

So, the equation of the line with a slope of 2 and passing through the point (3, 5) is y = 2x – 1 in slope-intercept form.

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.