When it comes to representing linear equations in a simple and elegant manner, the slope-intercept form plays a crucial role in the study of that sort of concept. The concept of slope-intercept form is a very useful concept in order to provide a comprehensible insight into the behavior of lines in a 2D plane.

The slope-intercept form is an effective and useful technique for various disciplines. It empowers us to analyze and apprehend the concept of correlations between unknowns (variables) because it can be applied easily i.e. easy to use in daily life.

The slope-intercept form simplifies the complex problems of graphing and clarifies the behavior of the linear equations in a concise and precise way. In this way, the concept of the slope-intercept form makes it easyto apprehend the terms of the slope and the y-intercept.

In this blog, we will discuss the concept of the slope-intercept form. We will unravel its components, and explore its examples.

What is the Slope-Intercept Form?

The slope-intercept form is a fundamental building block for interpreting linear equations and their graphical representations in mathematics. The slope-intercept form is a way of expressing a linear equation in the form of:

y = mx + b ———-(i)

Where

• yrepresents the dependent variable, which is represented on the vertical axis,
• x denotes the autonomous (independent)variable, which is representedon the horizontal axis,
• m represents the gradient (slope) of the line,
• bindicates the y-intercept i.e.the point (position) at which the line intersects the y-axis.

Note: Point-slope form (y – y₁ )= m (x – x₁) offers an alternative representation of linear equations. Converting to slope-intercept form allows for a more intuitive interpretation of the equation’s elements.

The Components of the Slope-Intercept Form:

Here we will elaborate on the core components of the slope-intercept form.

Understanding the Slope (m):

The gradient (slope), represented by m, signifies the steepness of a line i.e. inclines or declines. The slope of a line identifies and computes the steepness that a line has. It can also be calculated by the comparison of the 𝚫y and 𝚫x i.e. change in the vertical axis (y coordinate) and change in the horizontal axis (x coordinate) between two distinct points on the line.

Unveiling the Y-Intercept (b):

The y-intercept is the position or point where the x coordinate is 0and it is denoted by b. It provides a base for the trajectory (path) of the line.

Graphing Using Slope-Intercept Form:

Plotting the Y-Intercept:The slope-intercept form is used for graphing a line with plotting b (the y-intercept). It can be observed thatb is the position where the horizontal coordinate is zero i.e. x = 0. So, (0, b) is the coordinates of the point and bis the value of the y-intercept.

Grasping the Slope’s Influence:

The slope (m) is crucial in understanding the line’s behavior.

• A positive slope indicates an upward climb from left to right.
• A negative slope signifies a descent.
• The greater the slope’s absolute value, the steeper the line becomes.

Misconception to Avoid:

Mistakes in slope-intercept form often stem from errors in calculation or misunderstanding of the variables. Double-checking calculations and understanding the meaning of each variable can help avoid these errors.

How to evaluate equation of the line?

Here we will explore some important cases with examples of the slope-intercept form in detail.

Case 1. When Two Points are Given:

Example:

Find the equation of the straight line following the concept of slope-intercept form using points (6, 1), (8, 7)

Solution:

Step 1: Given data

x₁ = 6, x₂ = 8, y₁ = 1 and y₂ = 7

Step 2: Find the gradient (m) of the line.

m =  y/ 𝚫x = (y₂ – y₁)/ (x₂ – x₁)

m = (7 – 1)/ (8 – 6)

m = 6/ 2

m = 3

Step 3: Put the value of c i.e. y-intercept using the value of gradient (m = 3) and any one point in the slope-intercept formula.

y = mx + c

7 = (3) (8) + c             [using (8, 7)]

7 = 24 + c

7 – 24 = c

– 17 = c

c = – 17

Step 4:Put the values of slope (m) and intercept (c) in the slope-intercept formula.

y = mx + c

y = (3)x + (- 17)

y = 3x – 17 Ans.

Case 2. When Gradient and Y-Intercept are Given:

Example:

Find an equation of the straight line if the gradient is 5 and the y-intercept is – 3.

Solution:

Step 1: Given data

m = 5, c = – 3

Step 2:Use the slope-intercept formula to find out the equation of the straight line.

y = mx + c

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

y = (5)x + (- 3) = 5x -3 Ans.

Case 3: When Gradient and one Point are Given:

Example:

Find the equation of a straight line if m = – 3 and the point is (4, 2).

Solution:

Step 1:Given data

Here m = – 3, x = 4 and y = 2

Step 2:Find the value of c i.e. y-intercept using the point (4, 2) and the value of m = – 3 in the slope-intercept formula.

y = mx + c

2 = (- 3) (4) + c

2 = – 12 + c

2 + 12 = c

c = 14

Step 3:Put the values of m and c in the slope-intercept formula.

y = mx + c

y = (- 3)x + (14)

y = – 3x +14 Ans.

You can try AllMath’s slope intercept calculator to deal with the above cases of finding slope intercept form with steps.

Wrap Up:

In this article, we have addressed the concept of the slope-intercept form which is the core dimensional topic for the study of the equation of the straight line in algebra. We have elaborated on its definition and important components.

In the last section, we have solved some examples in order to apprehend the computations related to the slope-intercept form and equation of a straight line. Hopefully, by understanding this article, you will be able to tackle the problems about slope-intercept form, equation of the straight line, etc.