Basics of Slope-Intercept Form: Definition, Derivation, & Calculations


The slope-intercept form is an essential concept in algebra that plays a fundamental role in graphing linear equations, especially the equation of the straight line, precisely and concisely. The concept of the slope-intercept form is very useful and assists in analyzing data, finding valuable insights, making important predictions, and solving real-world problems.

This article will address the important concept of the slope-intercept form. We will discuss its definition and derivation of the slope-intercept form. We will also give some examples in order to apprehend the important concept of the slope-intercept form.


Slope-Intercept Form: Definition:


The slope-intercept form can be defined with the help of the following formula:


y = mx + b


Here m stands for the gradient (slope) and b denotes the y-intercept, and x and y are variables. The slope (m) indicates how steep the line is, a higher slope denotes a steeper inclination. While a negative slope denotes a falling slope, a positive slope implies an upward inclination.


Note: Only linear equations, which are represented on a graph by straight lines and whose graphs can only be depicted in two dimensions, i.e., on a plan, can be described using the slope-intercept form.


Derivation of Slope-Intercept Form:


Consider a line whose slope is m and whose y-intercept is b. This line touches the vertical axis (y-axis) at (0, c). Now, consider another point on the line given line at the position (x, y) as shown in the following Fig.


Suppose that (x1, y1) = (0, c) and (x2, y2) = (x, y). Now employing the slope-intercept formula,


gradient = (y2 – y1) / (x2 – x1)


gradient = (y – c) / (x – 0)


gradient = (y – c) / x


Multiplying on both sides with ‘x’, we will get:


mx = y – c


mx + c = y


Arranging the above equation:


y = mx + c


This is the general equation for a straight line that includes both the slope and the y-intercept. The slope-intercept form of the line's equation is what gives it that name. The slope intercept formula is obtained as a result.


Note: Once you encounter an equation in slope-intercept form, extracting the slope and y-intercept becomes a breeze.


How to find equation of line using slope intercept form?


Follow the below examples to learn how to find the line’s equation with the help of slope intercept form equation.


Example 1.


Derive the equation of the straight line if:


m = - 2/5, Point (1/2, - 3).


Solution:


Step 1: Given data:


Slope = - 2/5, x = 1/2 and y = - 3


Step 2: Calculate the value of c employing the point (1/2, - 3) and m = - 2/5 in the slope-intercept form:


y = mx + c


- 3 = (- 2/5) (1/2) + c


- 3 = - 1/5 + c


- 3 + 1/5 = c


c = - 14 / 5


Step 3: Place the relevant values of both m and c that we have


y = mx + c


y = (- 2/5)x + (- 14 / 5)


y = - 2x/5 – 14/5 Ans.


Example 2.


Derive the equation of a straight line if:


m = - 1/4, Point (4/5, - 1)


Solution:


Step 1: Given data:


m = - 1/4, Point (4/5, - 1)


Step 2: Compute the value of c employing the point (4/5, - 1) and m = - 1/4 in the slope-intercept form.


Y = mx + c


- 1 = (- 1/4) (4/5) + c


- 1 = - 1/5 + c


- 1 + 1/5 = c


c = - 4/5


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


y = mx + c


y = (- 1/4)x + (- 4/5)


y = - x/4 – 4/5 Ans.


Example 3:


Derive the equation of the straight line for the following given points:


(- 7, - 2) and (2, 7)


by employing the slope-intercept form.


Solution:


Step 1: Given data:


x1 = - 7, x2 = 2,


y1 = 2, y2 = 7


Step 2: To compute the m of the line.


m = (y2 – y1) / (x2 – x1)


m = (7 – (- 2)) / (2 – (- 7)


m = 7 + 2 / 2 + 7


m = 9 / 9


m = 1


Step 3: Find out the value of c by putting the point (2, 7)


y = mx + c


7 = (1) (2) + c


7 = 2 + c


7 – 2 = c


c = 5


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


y = m x + c


y = (1) x + 5 = x – 5 Ans.


Example 4:


Derive the equation of a straight line if:


m = - 1/3, c = - 2


Solution:


Step 1: Given data:


m = - 1/3, c = - 2


Step 2: Write down the slope-intercept formula.


y = mx + c


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


y = (- 1/3)x + (- 2)


y = - x/3 - 2


You can also obtain the result of a line's equation in term of slope intercept form in a fraction of a second by using a slope intercept equation calculator.


Wrap Up:


In this article, we have elaborated on the concept of the slope-intercept form. We have explored its definition, significant derivation, and some examples that will assist in understanding this concept precisely.


Hopefully, by reading this article you will be able to tackle the problems related to the slope-intercept form. It will also empower you to apprehend how to derive slope-intercept form.


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