Slope of a Line is the measure of the steepness of a line, a surface, or a curve, whichever is the point of consideration. The slope of a Line is a fundamental concept in the stream of calculus or coordinate geometry, or we can say the slope of a line is fundamental to the complete mathematics subject. The understanding of slope helps us solve many problems in mathematics, physics, or engineering.
In this article, we will learn about the slope of a line in detail, how to calculate the slope of a straight line with its various methods, and also the equation for the slope of a line.
What is a Slope?
In mathematics, the slope is the measure of the steepness of a line or a surface, which means the slope of a line tells us how steep a line or surface is inclined with the positive x-axis. The higher the slope of a curve or a line, the greater the amount of steepness it will show in the graph. Generally, the slope is defined in two-dimensional coordinates as the ratio of the change in the y-coordinate with respect to the change in the x-coordinate.
Similar to the general slope, the slope of a line or the slope of a straight line is the measure of the steepness of a line and is mathematically given as the ratio of the change in the y-coordinate to the change in the x-coordinate of the line. Other than this mathematical interpretation, the slope has a physical interpretation that tells us about the line that is in which direction the line will tilt.
A higher positive slope shows less tilt in the positive x direction, and a lower positive slope shows a higher tilt in the positive x direction. For the negative slope, the higher the absolute value of the negative slope higher the tilt towards the negative x direction.
The formula of the slope is given by the ratio of change in the y coordinate to the change in the x coordinate, which can be written mathematically as
Slope = Change in y coordinate/Change in x coordinate = Δy/Δx
Slope of a Line
If the inclination of the line with a positive x-axis is θ, then the slope is given as follows:
m = tan θ
Slope of a Line Equation
The slope of a line is given by the equation
y - y1 = m(x - x1)
⇒ y = mx + C where m is slope and C is intercept
How to Find Slope of a Line?
There are various methods to determine the slope of a line, which can be used appropriately on the basis of the given conditions. These methods are:
- Calculation of Slope between Two Points
- Calculation of Slope from Graph
- Calculation of Slope from Table
Calculation of Slope Between Two Points
If the given points are (x1, y1) and (x2, y2) then the slope of a line passing through both points is given by:
\bold{\text{Slope} = \frac{y_2 - y_1}{x_2 - x_1}}
Example: Find the point of the slope if points are (4, 2) and (8, 12).
Solution:
Given:
Point A (4,2) and point B (8,12)
Coordinate x1 and y1 is 4 and 2
Coordinate x2 and y2 is 8 and 12
Thus, m=\frac{(y_2-y_1)}{(x_2-x_1 )}
⇒ m = (12 - 2)/(8 - 4)
⇒ m = 10/4 = 2.5
Calculation of Slopes from Graph
Calculation of the slope from a Graph can be achieved using the following steps:
Step 1: Mark two points on the line with their coordinates.
Step 2: Use the Formula for the Slope between two points to calculate the slope.
Example: Find the slope of the following line in the graph.
Solution:
We have to find Δx and Δy (change in x and change in y)
So, change in Δx is 6 and change is Δy -3
Now slope m is given as follows:
m = Δy/Δx
⇒ m = 6/-3 = -2
Calculation of Slope from Table
Calculation of the Slope from the Table can be done using the following steps:
Step 1: Choose two values of x and its corresponding values of y from the table.
Step 2: Calculate the change in x value and change in y value.
Step 3: Calculate the slope using the formula, Slope = change in y-values/change in x-values.
Example 1: Calculate the slope between x = 1 and x = 3 of the following table.
x-value | y-value |
|---|
1 | 5 |
2 | 7 |
3 | 9 |
4 | 11 |
5 | 13 |
Solution:
Change in x-values = 3 - 1 = 2
Change in y-values = 9 - 5 = 4
As we know, Slope = change in y-values/change in x-values
⇒ Slope = 4/2 = 2
So, the slope between x=1 and x=3 in this table is 2.
Example 2: Calculate the slope of the following table.
Solution:
Identify change in each consecutive pair of y so change in the y is 5, 5 and 5.
Identify change in each consecutive pair of x so change in the x is 1, 1 and 1.
Now writing the ratio using slope formula 5/1, 5/1 and 5/1.
So, slope from table is 5.
Positive and Negative Slope
A line is said to have a positive slope if it is making less than the right angle with the positive x-axis and a line is said to have a negative slope if it makes more than the right angle with the positive x-axis.
In other words, a line with a positive slope looks tilted forward in the direction of the positive x-axis, and a line with a negative slope looks tilted backward in the direction of the negative y-axis.
Slopes of Different Lines
There can be various different lines that can be named such as:
- Horizontal Line
- Vertical Line
- Perpendicular Lines
- Parallel Lines
Let's discuss the Slope of these various different lines as follows:
Slope of Horizontal Line
The line that is parallel to the x-axis is called a horizontal line and the slope of a Horizontal Line is 0 as there is no change in the y-coordinate throughout the line for any change in the x-coordinate. Since the slope of the Horizontal Line is Zero, it is also called a Zero Slope line. Thus mathematically we can represent this as:
Slope of Horizontal Line = 0/change in x-coordinate = 0
Slope of Vertical Line
The line parallel to the y-axis is called a vertical line and the slope of a Vertical Line is not defined as there is no change in the x-coordinate throughout the line for any change in the y-coordinate. The Vertical Line is also called Undefined Slope Line. Thus mathematically we can represent this as:
Slope of Vertical Line = Change in y-coordinate/0 = Not Defined
Slope of Perpendicular Lines
The slope of Perpendicular Lines are inversely proportional to each other and their product is -1. In other words, if we have two lines with slopes m1 and m2, then the condition for those two lines to be perpendicular is:
m1 = -1/m2
OR
m1 × m2 = -1
Slope of Parallel Lines
The slope of Parallel Lines is the same as both the lines are at the same inclined with the positive x-axis. In other words, if the slope of one line is m then the slope of a line parallel to that line is also m.
The equation of a line in the slope-intercept form is given as follows:
y = mx + c
Where,
- m is the slope of the line
- c is the y-intercept cut by the line
Equation of Line Using Slope
If a line is passing through a point (x1, y1) and its slope is m, then the equation of a line is given as follows:
y - y1 = m(x - x1)
Where x and y represent all the coordinates of the line.
We can also write the same equation using the two points from which the line is passing. If the line passes through (x1, y1) and (x2, y2) then its equation is given by:
\bold{y - y_1 = \frac{y_2 - y_1}{x_2 - x_1}\cdot \left(x - x_1\right)}
Example 1: Find the equation of a line given in the graph.
Solution:
Slope of the graph is, m = 8/2 = 4
and we know the equation of line passing through (x1, y1) with slope m is given by
y – y1 = m (x – x1)
Thus, equation of line (4,2) with slope 4 is
y – 2 = 4 (x – 4)
⇒ y – 2 = 4x – 16
⇒ y = 4x – 16 +2
⇒ y = 4x – 14
Example 2: Find the equation of the line given in the graph.
Solution:
Two given points (x1, y1) and (x2, y2) are A (2,3) and B (5,7)
y-y_1= \frac{(y_2-y_1)}{(x_2-x_1)}\times (x-x_1 )
⇒ y-3= {(7 -3)/(5-2)} (x-2)
⇒ y-3= \frac{(7-3)}{(5-2)}\times (x-2)
⇒ y-3= \frac{(4)}{(3)}\times (x-2)
⇒ 3y-9 = 4x-8
⇒ 3y = 4x+1
Equation of the line whose length of the perpendicular from the origin is p and the angle made by the perpendicular with the positive x-axis is given by α is given by:
x cos α + y sin α = p
This is known as the normal form of the line.
In the case of the general form of the line Ax + By + C = 0 can be represented in normal form as:
\bold{A cos α + B sin α = – p}
From this we can say that \cos α = \frac{-p}{A} and \sin α = \frac{-p}{B}
Also, it can be inferred that,
cos2α + sin2α = \frac{(p)}{(A)}^{2}+ \frac{(p)}{(B)}^{2}
⇒ 1 = p^{2} \frac{(A^2 + B^2)}{(A^2. B^2)}
⇒ p=\frac{A.B}{\sqrt{A^{2} + B^{2}}}
From the general equation of a straight-line Ax + By + C = 0, we can conclude the following:
- Slope is given by -A/B, given that B ≠ 0.
- x-intercept is given by -C/A and the y-intercept is given by -C/B.
- It can be seen from the above discussion that:
- p=±\frac{A.B}{\sqrt{A^{2} + B^{2}}}
- cosα=±\frac{B}{\sqrt{A^{2} + B^{2}}}
- sinα=±\frac{A}{\sqrt{A^{2} + B^{2}}}
- If two points (x1, y1) and (x2, y2) are said to lie on the same side of the line Ax + By + C = 0, then the expressions Ax1+ By1 + C and Ax2 + By2 + C will have the same sign or else these points would lie on the opposite sides of the line.
Angle between Two Lines
When two lines with slopes m1 and m2 intersect they form angles. The relation between the slope of the intersecting lines and the angle so formed is given as follows:
tanθ = |\frac{m_2-m_1}{1+m_2*m_1}|
if |\frac{m_2-m_1}{1+m_2*m_1}| > 0, acute angle is formed between the lines
if |\frac{m_2-m_1}{1+m_2*m_1}| < 0, obtuse angle is formed between the lines
Read More,
Sample Problems on Slope of a Line
Problem 1: Find the slope of points (1,2) and (2,3).
Solution:
As slope is given as m = (y2 - y1)/(x2 - x1)
⇒ m = (3 - 2)/(2 - 1)
⇒ m = 1
Problem 2: Find the value of x if the slope is 2 and points are (2,2) and (x,6).
Solution:
m = (y2 - y1)/(x2 - x1)
⇒ 2 = (6 - 2)/(x - 2)
⇒ 4 = 2(x-2)
⇒ x-2 = 2
⇒ x = 4
Problem 3: Find the value of y if slope is 3 and points are (2,13) and (4, y).
Solution:
m = (y2 - y1)/(x2 - x1)
⇒ 3 = (y - 13)/(4 - 2)
⇒ y - 13 = 3(2)
⇒ y - 13 = 6
⇒ y = 6 + 13 = 19
Problem 4: Find the line passing from coordinates (2,5) and the slope of a line is 5.
Solution:
Slope m = 4
y – y1= m (x – x1)
We know slope m = 5 and point (x1, y1) = (2,5)
Now putting these value in equation
⇒ y – 5 = 5 x (x – 2)
⇒ y – 5 = 5x – 10
⇒ y = 5x – 10 + 5
⇒ y = 5x -5
Practice Questions of Slope of a Line
Problem 1: Find the slope of points (3, 4) and (5, 7).
Problem 2: Find the value of x if the slope is -1 and the points are (x, 4) and (2, 6).
Problem 3: Find the value of y if the slope is 2 and the points are (1, y) and (3, 8).
Problem 4: Find the equation of the line passing through the coordinates (4, 6) with a slope of -2.
Similar Reads
CBSE Class 11 Maths Notes CBSE Class 11 Maths Revision Notes have been designed in the most basic and detailed format possible, covering nearly all domains such as differential calculus, arithmetic, trigonometry, and coordinate geometry. We know how hard it gets when you shift to an altogether new grade where subjects are no
15+ min read
Chapter 1: Sets
Representation of a SetSets are defined as collections of well-defined data. In Math, a Set is a tool that helps to classify and collect data belonging to the same category. Even though the elements used in sets are all different from each other, they are all similar as they belong to one group. For instance, a set of dif
8 min read
Types Of SetsIn mathematics, a set is defined as a well-defined collection of distinct elements that share a common property. These elements— like numbers, letters, or even other sets are listed in curly brackets "{ }" and represented by capital letters. For example, a set can include days of the week. The diffe
13 min read
Universal SetsUniversal Set is a set that has all the elements associated with a given set, without any repetition. Suppose we have two sets P = {1, 3, 5} and Q = {2, 4, 6} then the universal set of P and Q is U = {1, 2, 3, 4, 5, 6}. We generally use U to denote universal sets. Universal Set is a type of set that
6 min read
Venn DiagramVenn diagrams are visual tools used to show relationships between different sets. They use overlapping circles to represent how sets intersect, share elements, or stay separate. These diagrams help categorize items, making it easier to understand similarities and differences. In mathematics, Venn di
14 min read
Operations on SetsSets are fundamental in mathematics and are collections of distinct objects, considered as a whole. In this article, we will explore the basic operations you can perform on sets, such as union, intersection, difference, and complement. These operations help us understand how sets interact with each
15+ min read
Union of SetsUnion of two sets means finding a set containing all the values in both sets. It is denoted using the symbol '∪' and is read as the union. Example 1:If A = {1, 3. 5. 7} and B = {1, 2, 3} then A∪B is read as A union B and its value is,A∪B = {1, 2, 3, 5, 7}Example 2:If A = {1, 3. 5.7} and B = {2, 4} t
12 min read
Chapter 2: Relations & Functions
Cartesian Product of SetsThe term 'product' mathematically refers to the result obtained when two or more values are multiplied together. For example, 45 is the product of 9 and 5.To understand the Cartesian product of sets, one must first be familiar with basic set operations such as union and intersection, which are appli
7 min read
Relations and FunctionsIn mathematics, we often deal with sets of numbers or objects and the ways they are connected. Two important concepts that help us describe these connections are relations and functions.A relation is simply a connection between two sets of objects. Think of it as a rule that pairs elements from one
3 min read
Domain and Range of RelationsThe domain is the set of all possible input values (the "x" values), and the range is the set of all possible output values (the "y" values) in a relation.For any two non-empty sets A and B, we define the relation R as the subset of the Cartesian product of A × B where each member of set A is relate
8 min read
Piecewise FunctionPiecewise Function is a function that behaves differently for different types of input. As we know a function is a mathematical object which associates each input with exactly one output. For example: If a function takes on any input and gives the output as 3. It can be represented mathematically as
11 min read
Range of a FunctionFunctions in math can be thought of as vending machines. Given the money in the form of input, they give some cans or cookies in return. Similarly, functions take some input numbers and give us some output. It can be said that, in real life, Everything can be formulated and solved with the help of f
7 min read
Chapter 3: Trigonometric Functions
Chapter 4: Principle of Mathematical Induction
Chapter 5: Complex Numbers and Quadratic Equations
Complex NumbersComplex numbers are an essential concept in mathematics, extending the idea of numbers to include solutions for equations that don't have real solutions. Complex numbers have applications in many scientific research areas, signal processing, electromagnetism, fluid dynamics, quantum mechanics, and v
12 min read
Algebra of Real FunctionsThe algebra of real functions refers to the study and application of algebraic operations on functions that map real numbers to real numbers. A function can be thought of as a rule or set of rules which map an input to an output knows as its image. It is represented as x ⇢ Function ⇢ y. A real funct
5 min read
Algebraic Operations on Complex NumbersA complex number is a number that includes both a real and an imaginary part. It is written in the form:z = a + biWhere:a is the real part,b is the imaginary part,i is the imaginary unit, satisfying i2 = −1.Algebraic operations on complex numbers follow specific rules based on their real and imagina
7 min read
Polar Representation of Complex NumbersComplex numbers, which take the form z = x + yi, can also be represented in a way that highlights their geometric properties. This alternative representation is known as the polar form. The polar representation of a complex number expresses it in terms of its magnitude (modulus) and direction (argum
9 min read
Absolute Value of a Complex NumberThe absolute value (also called the modulus) of a complex number z = a + bi is its distance from the origin in the complex plane. The absolute value tells you how far a number is from zero, regardless of its direction (positive or negative).It is denoted as ∣z∣ and is given by the formula:|z| = \sqr
7 min read
Conjugate of Complex NumbersIn the world of mathematics, complex numbers are one of the most important discoveries by mathematicians as they help us solve many real-life problems in various fields such as the study of electromagnetic waves, engineering, and physics.The Conjugate of a Complex Number is also a complex number obt
6 min read
Imaginary NumbersImaginary numbers are numbers as the name suggests are the number that is not real numbers. All the numbers real and imaginary come under the categories of complex numbers. Imaginary numbers are very useful in solving quadratic equations and other equations whose solutions can not easily be found us
9 min read
Chapter 6: Linear Inequalities
Compound InequalitiesCompound Inequalities are the combination of two or more inequalities. These inequalities are combined using two conditions that are AND, and OR. These conditions have specific meanings and they are solved differently. The inequities in compound inequalities are individually solved using normal rule
10 min read
Algebraic Solutions of Linear Inequalities in One VariableA linear inequality is a mathematical expression involving an inequality symbol (<, >, ≤, or ≥) and a linear expression. Unlike linear equations, which give a specific solution, linear inequalities define a range of possible solutions.Example: 2x+3>5 In this case, the inequality indicates t
8 min read
Graphical Solution of Linear Inequalities in Two VariablesWe know how to formulate equations of different degree, and it is used a lot in real life, but the question arises, is it always possible to convert a situation into an equation? Sometimes we get statements like, the number of Covid cases per day in Delhi has reached more than 10,000. This phrase “L
8 min read
Solving Linear Inequalities Word ProblemsWe are well versed with equations in multiple variables. Linear Equations represent a point in a single dimension, a line in a two-dimensional, and a plane in a three-dimensional world. Solutions to linear inequalities represent a region of the Cartesian plane. It becomes essential for us to know ho
10 min read
Chapter 7: Permutations and Combinations
Fundamental Principle of CountingThe fundamental principle of counting is a basic concept used to determine the total number of possible outcomes in a situation where there are multiple independent events. It allows us to count a large number of possibilities without needing to list each one individually.For example, consider guess
11 min read
PermutationIn Mathematics, Permutation is defined as a mathematical concept that determines the number of possible arrangements for a specific set of elements. therefore, it plays a big role in computer science, cryptography, and operations research. For example, take a set {1, 2, 3}:All Permutations taking al
15+ min read
CombinationsCombination is a way of choosing items from a set, (unlike permutations) when the order of selection doesn't matter. In smaller cases, it's possible to count the number of combinations. Combination refers to the mixture of n things taken k at a time without repetition.Example: For set S = {a, b, c},
8 min read
Chapter 8: Binomial Theorem
Chapter 9: Sequences and Series
Sequences and SeriesA sequence is an ordered list of numbers following a specific rule. Each number in a sequence is called a "term." The order in which terms are arranged is crucial, as each term has a specific position, often denoted as an​, where n indicates the position in the sequence.For example:2, 5, 8, 11, 14,
10 min read
General and Middle Terms - Binomial Theorem - Class 11 MathsBinomial theorem or expansion describes the algebraic expansion of powers of a binomial. According to this theorem, it is possible to expand the polynomial "(a + b)n" into a sum involving terms of the form "axzyc", the exponents z and c are non-negative integers where z + c = n, and the coefficient
7 min read
Arithmetic SeriesAn arithmetic series is the sum of the terms of an arithmetic sequence, where an arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. Or we can say that an arithmetic progression can be defined as a sequence of numbers in which for every pair of
5 min read
Arithmetic SequenceAn arithmetic sequence or progression is defined as a sequence of numbers in which the difference between one term and the next term remains constant.For example: the given below sequence has a common difference of 1.1 2 3 4 5 . . . n ⇑ ⇑ ⇑ ⇑ ⇑ . . . 1st 2nd 3rd 4th 5th . . . nth TermsThe Arithmetic
8 min read
Geometric Progression or GPGeometric Progression (GP) is a sequence of numbers where each term after the first is found by multiplying the previous term by a constant called the common ratio.For Example, the sequence given below forms a GP with a common ratio of 2 1 2 4 8 16 . . . n⇑ ⇑ ⇑ ⇑ ⇑ . . . 1st 2nd 3rd 4th 5th . . . nt
12 min read
Geometric SeriesIn a Geometric Series, every next term is the multiplication of its Previous term by a certain constant, and depending upon the value of the constant, the Series may increase or decrease.Geometric Sequence is given as: a, ar, ar2, ar3, ar4,..... {Infinite Sequence}a, ar, ar2, ar3, ar4, ....... arn {
3 min read
Arithmetic Progression and Geometric ProgressionArithmetic Progression and Geometric Progression: The word "sequence" in English means a collection of some numbers or objects in such a way that it has a first member, a second member, and so on. Sequences can be of anything, for example. - January, February, .... is the sequence of months in a yea
10 min read
Special Series in Maths - Sequences and Series | Class 11 MathsSpecial Series: A series can be defined as the sum of all the numbers of the given sequence. The sequences are finite as well as infinite. In the same way, the series can also be finite or infinite. For example, consider a sequence as 1, 3, 5, 7, … Then the series of these terms will be 1 + 3 + 5 +
10 min read