The slope-intercept form is a significant subject matter in the study of linear equations especially when working with the equation of the straight line, which is the core dimension, used in algebraic expressions.

It describes the linear relationship b/w the variables (independent and dependent) to estimate the rate of change. In this article, we will elaborate on slope intercept form, its definition, and formula, and at last, we will solve some examples.

Defining the Slope-Intercept Form:

The slope-intercept form is a technique for figuring out the equation of a straight line. To compute the equation of a straight line, we need some information about the slope (which is also termed as Gradient sometimes), a point where the line touches the y-axis.

Slope computes the steepness or inclination of the line and Intercept points out the position in the coordinate system where the line touches a coordinate axis. The slope intercept form is very useful for working out problems involving lines and graphs.

Before introducing its formula, it is necessary to note that a rectangular or Cartesian plane comprises two straight lines bisecting each other at an angle of 90^0, and their point of intersection is called the origin denoted 0. One is a horizontal line known as the x-axis and the second is a vertical line known as the y-axis respectively.

Formula:

Mathematically, the slope-intercept form can be defined as

y = mx + c

In the above equation,

• y is the dependent variable
• m is the slope of the line
• x is an unknown that is independent
• c is the y-intercept (A point where the line touches the y-axis)

Where,

Gradient = m = 𝚫y/ 𝚫x = (y₂ – y₁)/ (x₂ – x₁)

We can also comprehend that slope is rise (change along the vertical axis) over run (change along the x-axis). Some important facts about the slope of a line, are given below:

• Gradient will be 0 if the straight line is horizontal i.e. parallel to the x-axis.
• Gradient will be undefined if the straight line is vertical i.e. parallel to the y-axis.
• Gradient is positive if 𝚫y and 𝚫x have the same signs.
• Gradient is negative if 𝚫y and 𝚫x have opposite signs.
• If m > 0, it implies both variables x and y are directly proportional to each other.
• If m < 0, then both variables x and y are inversely proportional to each other.
• If c > 0, then the y-intercept will be above the origin on the y-axis.
• If c < 0, then the y-intercept will be below the origin on the y-axis.

Cases of slope-intercept Form:

Since we have already discussed this in detail here, finding the equation of a straight line requires the slope-intercept form. Now we will discuss here some useful techniques to find the equation of a straight line using the concept of slope-intercept form explaining some cases with the help of examples.

• When gradient and y-intercept are given
• When two pints are given
• When gradient and one point are given

Below is a manual method to find the slope-intercept form according to the given cases.

 Case 1: When Two Points Are Given:

Example 1:

Find out the equation of the straight line for the following given points:

(- 2, – 3) and (5, 7)

by employing the slope-intercept form.

Solution:

Step 1: Given data:

x₁ = – 2, x₂ = 5,

y₁ = – 3, y₂ = 7

Step 2:  Calculate the m of the line.

m = (y₂ – y₁) / (x₂ – x₁)

m = (7 – (- 3)) / (5 – (- 2)

m = 7 + 3 / 5 + 2

m = 10 / 7

Step 3: Find out the value of c by putting the point (5, 7)

y = mx + c

7 = (10/7) (5) + c             [using (2, 3)]

7 = 50/7 + c

7 – 50/7 = c

c = – 1/7

Step 4: Put the values of m and c in the given formula.

y = m x + c

y = (10/7) x + (-1/7) = 10x/7 – 1/7

y = 1.43x – 0.14

A slope intercept form calculator by AllMath is an alternate way to solve the problems of finding line equation through slope intercept form.

Solved by Slope intercept calculator by AllMath

Case 2: When gradient and one point are given:

Example 1.

What will be the equation of the straight line if:

m = – 4, Point (3, – 1).

Solution:

Step 1: Given data:

Slope = – 4, x = 3 and y = – 1

Step 2: Calculate the value of c employing the point (3, -1) and m = – 4 in the slope-intercept form:

y = mx + c

– 1 = (- 4) (3) + c

– 1 = – 12 + c

1 + 12 = c

c = 13

Step 3: Place the relevant values of both m and c.

y = mx + c

y = (- 4)x + (13)

y = – 4x + 13 Ans.

Example 2.

What will be the equation of a straight line if:

m = – 1, Point (2, – 3)

Solution:

Step 1: Given data:

m = – 1, Point (2, – 3)

Step 2: Calculate the value of c employing the point (2, – 3) and m = – 1 in the slope-intercept form.

Y = mx + c

– 3 = (-1) (2) + c

– 3 = – 2 + c

– 3 + 2 = c

c = – 1

Step 3: Put the values both of the m and c.

y = mx + c

y = (- 1)x + (- 1)

y = – x – 1 Ans.

Case 3: When Slope and Y-intercept are Given

Example 1.

What will be the equation of a straight line if:

m =  3/2, c = – 2

Solution:

Step 1: Given data.

m =  3/2, c = – 2

Step 2: Place the relevant values in the above formula and simplify.

y = (3/2)x + (- 2)

y = 3x/2 – 2 Ans.

Example 2.

What will be the equation of a straight line if:

m = 3/4, c = – 5

Solution:

Step 1: Given data:

m = 3/4, c = – 5

Step 2: Write down the slope-intercept formula.

y = mx + c

Step 3: Place the relevant values in the formula and simplify.

y = (3/4)x + (- 5)

y = 3x/4 – 5 Ans.

Conclusion:

In this article, we have addressed an important concept of the slope-intercept form. We elaborated on its definition, formula, and calculation cases with the help of some solved examples. I hope that by reading and apprehending this article you will be able to tackle the problems related to slope-intercept form and linear equations.