Category: Maths Notes PPT

Relation and function both are closely related to each other, and to have a clear understanding of them, one must take proper knowledge from the maths experts on our website.

Relation

A relation is nothing but the connection of two sets by any means. In mathematics, it is a collection of ordered pairs that contain elements from one set to the other set.

A relation is a collection of ordered pairs, which contains an object from one set to the other set. For instance, X and Y are two sets, and ‘a’ is the object from set X and b is the object from set Y, then we can say that the objects are related to each other if the order pairs of (a, b) are in relation. 

Consider two arbitrary sets X and Y. The set of all ordered pairs (x,y) where x∈X and y∈Y is called the Cartesian product of X and Y. The product is designated as, read as X cross Y. By definition,

X х Y = {(x, y)} I x ∈ X and y ∈ Y} 

X х Y ≠ Y х X . The Cartesian product deals with ordered pairs, so the order in which the sets are considered is important. Using n(A) for the number of elements in a set A, we have:

n(X х Y) = n(X) х n(Y)

Function

It is a relation that defines the set of inputs to the set of outputs.

Note that all functions are relations, but not all relations are functions.

The relation that defines the set of input elements to the set of output elements is called a function. Each input element in the set X has exactly one output element in the set Y in a function. A function requires two conditions to be satisfied to qualify as a function:

1. Every x∈X must be related to y∈Y, i.e., the domain of f must be X and not a subset of X.

2. There is a requirement of uniqueness, which can be expressed as:

(x,y) ∈f and (x,z) ∈f ⇒ y = z

Sometimes we represent the function with a diagram: f : A⟶B or Af⟶B    

Functions are sometimes also called mappings or transformations.

Difference between Function and Relation in Maths

A relation from a set X to a set Y is any subset of the Cartesian product X×Y. 

A relation X to Y is a subset of X Y.

Let there be an X set and a Y set. An ordered pair (x,y) is called a relation in x and y. The first element in an ordered pair is called the domain, and the set of second elements is called the range of the relation.

Let us consider R as a relation from X to Y. Then R is a set of ordered pairs where each first element is taken from X and each second element is taken from Y. That is, for each x ∈ X and y ∈ Y, follows exactly one of the following:

1. x, y R; then“x is R-related to y”, written as xRy.

2. x, y ∉ R; then “x is not R-related to y”, written as xRy

If R is a relation from a set X to itself, that is, if R is a subset of X2 =X X, we say that R is a relation on X.

The Difference between a Relation and a Function

To understand the difference between a relationship that is a function and a relation that is not a function. All functions are relations, but not all relations are functions. The difference between a relation and a function is that a relationship can have many outputs for a single input, but a function has a single input for a single output. This is the basic factor to differentiate between relation and function. Relations are used, so those model concepts are formed. Relations give a sense of meaning like “greater than,” “is equal to,” or even “divides.”

A Relation is a group of ordered pairs of elements. It can be a subset of the Cartesian product. It is a dyadic relation or a two-place relation. Relations are used, so those model concepts are formed. Relations give a sense of meaning like “greater than,” “is equal to,” or even “divides.”

Function pertains to an ordered triple set consisting of X, Y, F. X, where X is the domain, Y is the co-domain, and F is the set of ordered pairs in both “a” and “b.” Each ordered pair contains a primary element from the “A” set. The second element comes from the co-domain, and it goes along with the necessary condition.

In a set B, it pertains to the image of the function. The domain and co-domain are both sets of real numbers. It doesn’t have to be the entire co-domain. It can be known as the range. Relations show the properties of items. In a way, some things can be linked in some way, so that’s why it’s called “relation.” It doesn’t imply that there are no in-betweens that can distinguish between relation and function. 

One thing good about it is the binary relation. It has all three sets. It includes the X, Y, and G. X and Y are arbitrary classes, and the “G” would have to be the subset of the Cartesian product, X x Y. They are known as the domain set of departure or even co-domain. G would be understood as a graph.

The function can be an item that takes a mixture of two-argument values that can give a single outcome. There is another difference between relation and function. The function should have a domain that results from the Cartesian product of two or more sets but is not necessary for relations.

Similarities between Logarithmic and Exponential Functions

1. Log functions can be written as exponential functions.

2. Logs of products involve addition and products of exponentials involve addition.

Benefits of Knowing the Difference

Other than learning the topics, students have to understand the difference between these topics. When you know the difference, it becomes easy to break down the seeds of knowledge and gain the consciousness of tiny topics related to it.

Our subject matter experts offer you a detailed explanation of the topic, Relation and Function, in the online maths class. You can join the maths online class to know more about the relation and function.

Differentiation and Integration are two mathematical concepts that primarily form Calculus and can be understood as two concepts opposite to each other. Differentiation in calculus is defined as the instantaneous rate of change of a function with respect to one of its variables. Leibniz’s notation is commonly used for differentiation, but some other notations like Euler’s and Lagrange’s notation are also commonly used.

Leibniz’s Notation – dy/dx is defined as the infinitesimal change in y due to an infinitesimal change dx in the value of x.

Euler’s Notation – D(y) or D[f(x)], where D is the differential operator.

Lagrange’s Notation – f'(x) s also termed as prime notation.

What is Differentiation in Maths?

The mathematical definition of differentiation is the change in the value of the function due to the change in the independent variable.

y=f(x)

Where y is a function of x. Any change in the value of y due to the change in the value of x is given by:

dy/dx

Solved Example: Differentiate y=4x2 + x – 4 w.r.t x

Ans: Differentiating the given equation y w.r.t x.

dy/dx=4.2.x +1-0

dy/dx=8x+1

Rules for Important Differentiation Formulas

Given below are the commonly used differentiation formulas. Please note that these are directly applicable formulas. Students are advised to remember these by heart. Let y be the function and dy/dx the derivative of the function.

• y = tan x, dy/dx = sec2x

• y = sin x, dy/dx = cos x

• y = cos x ,dy/dx = -sin x

• y = an, where a is any integer or fraction, dy/dx = nan-1

• y = ex(exponential function), dy/dx=ex

• y = ln(x), dy/dx = 1/x

• y = k, where k is any constant, dy/dx = 0

Rules for Compound Functions

Some of the basic rules that need to be followed while solving and differential problem are:

If the function whose derivative is to be calculated is the sum or difference of two functions, then the derivative of the function is the sum or difference of the individual derivative of the two functions.

F(x) = f(x) ± g(x)

F’(x) =f’(x) ± g’(x)

If the function in a question is the product of two individual functions, the derivative of the function is given by:

F(x) = f(x) x g(x)

F’(x) = f’(x) x g(x) + f(x) x g’(x)

If the function whose derivative is to be calculated is the division of two functions, the derivative is calculated as follows:

F(x) = f(x)/g(x)

F’(x) = [f’(x) x g(x) – f(x) x g’(x)]/g(x)2

Sometimes to solve complex functions, a substitution method is used. If a function y = f(x) = g(z) and if z = h(x), then:

dy/dx = dy/dzdz/dx

Applications of Differentiation in Real Life Problems

Differentiation has real-life applications. Differentiation is defined as the change in any quantity with respect to change in other quantities. Some real-life applications are:

• Acceleration is defined as the change of velocity with the change in time.

• Tangent and normal to a curve is derived using a derivative function.

• Maximum and minimum points of a graph to be used in the study of businesses etc.

• Also used to determine any temperature variation.

In Maths, often there are functions f(x) that are not continuous at a point of its domain D. These non-continuous functions are called a point of discontinuity of the function. In other words, in a graph, if the functions are not connected to each other they will be called a discontinuous function.

The discontinuity can be because of the following situations: 

1. If both the right-hand limit as well as the left-hand limit or maybe any one of them do not exist.

2. If both of them i.e., the right-hand limit and the left-hand limit of a function do exist but are not equal. 

3. If either of the two or maybe both are not equal to the function f(x). 

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We can see that in the graph given above, the limits of the function towards the left and the limits of the function towards the right are unequal so the limit at x = 3 does not exist anymore. These types of functions are said to be a discontinuity of a function.

Types of Discontinuity

In this flow chart of the types of discontinuity, we can see that there are two types of discontinuity i.e., removable discontinuity and non-removable discontinuity. Removable discontinuity has two parts i.e., missing point and isolated point. Non-removable discontinuity has three parts i.e., finite type, infinite type, and oscillatory discontinuity. 

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What is a  Removable Discontinuity?

We can call a discontinuity “removable discontinuity” if the limit of the function exists but either they are not equal to the function or they are not defined. However, there is a possibility of redefining a function in a way that the limit will be equal to the value of the function at a particular point. 

Missing Point Discontinuity

A missing discontinuity arises when the limit of the function exists at a point but the function is undefined at that point. In a graph, it is represented as an open circle at the point where it is left undefined.  

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Isolated Point Discontinuity

In an isolated point discontinuity, the limits of a function do not only exist but are also defined at a point. This does not mean that they both are not equal. 

What is a Non – Removable Discontinuity?

If the limit of a function does not exist then we call it a non-removable discontinuity. It is not possible to redefine a function to make it continuous. A non-removable discontinuity can further be divided into 3 parts i.e., a finite type of a discontinuity, an infinite type of a discontinuity, and an oscillatory discontinuity. 

Finite Type 

In a finite type of discontinuity, both the left as well as the right-hand limits do exist but they are unequal. In other words, it is when a two-sided limit does not exist, but both the two one-sided limits are finite yet they are not equal to each other. In a graph of a finite type of a discontinuity, the function will be represented as a vertical gap between the two branches of the function. When there is a non-negative difference between the two limits, we call it the Jump of Discontinuity. 

Infinite Type

We can say it is an infinite discontinuity if either one or both the Right-Hand and the Left-Hand Limit do not exist or they are Infinite. We also call it Essential Discontinuity. If a graph of a function has the line x = k, as a vertical asymptote, then the function becomes either positively or negatively infinite. Therefore, the function f(x) will be called as an infinite discontinuity.

Oscillatory Discontinuity

The oscillatory discontinuity is a discontinuity when the limits oscillate between any two finite quantities.

Solved Examples

Question 1) Solve the discontinuity of a function algebraically and graph it. 

[f(x) = frac{(x – 2)(x + 2)(x – 1)}{(x – 1)}]

Solution 1) We can remove or cancel the factor x = 1 from the numerator as well as the denominator. Therefore, we will be left with f(x) = (x – 2)(x + 1). since x = 1 is canceled, we get a removable discontinuity at x = 1. The graph will be represented as y = (x – 2)(x + 1) and a hole at x = 1. While graphing, y = (x – 2)(x + 1) as usual along with the hole.

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Question 2) Describe whether a function [f(x) = sinfrac{1}{x}] as continuity or discontinuity.

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Solution 2) The function is an oscillate infinitely as x approaches 0. The graph neither has a hole or a jump discontinuity nor does it shoot to infinity. However, it is also not continuous at x = 0.     

Question 3) Describe the discontinuity of the function from the graph given below:

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Solution 3) We can see that there are two jump discontinuity at x = -2 and x = 4. There is a removable discontinuity at x = 2 and an infinite discontinuity at x = 0.

Divisibility rules are basically to solve problems related to integer division in a very easy way. The divisibility rule has come to check whether the dividend integer can be completely divided by any other divisor integer or not.

In order to check the divisibility of a large number by interest will take about time. That’s why counter-time divisibility rules were introduced. So, in this article, we are going to discuss divisibility rules for 13. 

Apart from 13, there are divisibility rules for 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, and so on.

For Example:

Divisibility by 4 rule, 48 in a number which is completely divided by 4 as the sum of the last two digits of the number is divided by 4. Let’s find another number 47, which is not divisible by 4 as the sum of the last two digits of the number is not completely divided by 4. Using this simple rule, we can find if any number is divisible by 4 or not. 

Now, let’s discuss the divisibility rule for 13, using definitions and examples.

Different Divisibility Rules For 13

We have to read 4 different types of divisibility rules for 13. Let us explain to you with examples one by one. 

Divisibility Rule 1: 

For a given number, form alternating sums of blocks of three numbers from the right and move towards the left. Suppose (n1, n2, n3, n4, n5, n6…..)  is a number N, then if the number formed by the alternative sum of blocks of 3-3 digits from right to left (n1, n2, n3, –  n4, n5, n6, + …. ) is divisible by 13, then the number N is additionally divisible by 13.

Example: Let a number is 2,453,674. Find out whether it is divisible by 13 or not.

Solution: By applying Rule 1,674 – 453 + 2 = 223 is not divisible by 13 

Therefore, 2,453,674 also is not divisible by 13

Divisibility Rule 2:

If a number N is given, then multiply the last digit of N with 4 and add it to the rest truncate of the number. If the result is divisible by 13, then the number N is additionally divisible by 13.

Example: Let a number be 780. Find whether it is divisible by 13.

Solution: By applying Rule 2,780: 78 + 0 x 4 = 78 and, number 78 is divisible by 13 and gives divisor as 6. 

Therefore, 780 is also divisible by 13.

Divisibility Rule 3:

For a number N, to check whether it is divisible by 13 or not, subtract the last 2 digits of the number N from the 4 times multiple of the rest of the number.

Example: Let a number is 728. Check whether it is divisible by 13 or not.

Solution: By implementing the divisibility rule of 13, we get,2197: 21 x 4 – 97 = 97 – 84 = 13, and number 13 is divisible by 13, giving the result as 0.

Divisibility Rule 4: 

Multiply the last digit by 9 of a number N and subtract it from the rest of the number. If the result is divisible by 13, then the numeral N is also divisible by 13.

Example: If a number is 858 then find out whether it is divisible by 13 or not. 

Solution: By applying rule 4,936: 93 – 6 x 9 = 39, and 39 is divisible by 13Therefore, 936 is divisible by 13.

Questions:

Question 1.

(a).  Is 298 divisible by 13? 

Ans.    

Four times of the last digit  = 4 x 8 = 32             

Remaining left 29             

Addition = 29 + 32 = 61             

Since 61 is not divisible by 13             

∴  298 is not divisible by 13.

(b).  Is 247 Divisible by 13?

Ans. Four times of the last digit  = 4  x 7 = 28            

Remaining left 24            

Addition = 24 + 28 = 52            

 Since 52 is divisible by 13             

∴  247 is divisible by 13.

(c).  Is 317 Divisible by 13? 

Ans.  Four times of the last digit  = 4 x 7 = 28            

Remaining left 31            

Now, Addition = 28 + 31 = 59            

Since 59 is not divisible by 13                   

∴317 is not divisible by 13.

(d). Is 50661 Divisible by 13? 

Ans.   

Four times of the last digit = 4  x 1 = 4            

Remaining left 5066            

Now, Addition = 5066 + 4 = 5070            

Again, Four times of the last digit = 4  x 0 = 0            

Now, Addition = 507 + 0 = 507            

Again, Four times of the last digit = 4 x 7 = 28            

Now,  Addition = 50 + 28 = 78                

And, 78 is divisible by 13 at 13 x 6.

The concept of divisibility rules is introduced in the third grade. First, the students are taught about BODMAS. The concept of multiplication and division is elaborated in the higher classes. When students enter fourth grade they are taught about factors and multiples and prime factorization and composite numbers and integers. All these concepts form the basic foundation of mathematics and it is extremely important to have a good hold over these topics in order to get a higher grade in the class and it is only after understanding these topics that a student can go for higher studies.

The expert mathematical team at has curated all the study material related to divisibility rules and this article mainly deals with divisibility rules for 13. Students can find many other divisibility rules for different numbers as per their needs. This article explains in-depth what the divisibility rule is and why we use these rules. In order to get a concrete hold over the concept ’s team has also provided practice questions along with their solutions so that students can learn and understand in an easier way.

Why study divisibility rules?

These divisibility rules are tricks in mathematics that help students to play, manipulate numbers and also help in solving the problems in an easier and more efficient way. These divisibility rules help in improving mathematical skills. In order to use the divisibility rules, students should get a good hold over the divisibility concept before they can use these tricks or divisibility rules in their favor. Students should fully understand when and how to use these rules. The rules of divisibility are extremely important in the study of mathematics as it makes problem-solving easier. Students should be equipped with this concept as early as possible as it will only make them mentally strong and therefore will enable them to get an edge over the others.

These rules help students to build a number sense and help them to deal effectively with large numbers that usually scare students. The goal is to help students be familiar with the numbers and be flexible while working with them.

Divisibility rules help you to understand if a number is divisible by a certain number for example 3, 2, 4, etc. without actually using division or a Calculator.

The best possible way to learn divisibility rules is by repeatedly using them every day during homework or in the classroom as this will help in getting the concept and repeatedly using it is the only possible way by which a student can effectively learn these divisibility rules.

Matrix is one of the most powerful tools in mathematics. In simple words, it is a rectangular array of numbers organized in rows and columns. The number of rows and columns in a matrix determines its order or dimension. The general representation of the order of a matrix or array is m X n, where n represents the number of columns, while m represents the number of rows. The following is an example of a matrix or array.

[begin{bmatrix} 1 & 3 &2 \ 6 &2 &7 \ 3 &4 & 7 end{bmatrix}]

The above matrix has three rows and three columns. Hence, the order of this array is 3 X 3. There are many operations that you can perform on a matrix, which are known as transformations. Now, let’s look at the Elementary Operation of Matrix in detail in the article below.

Types of Elementary Operations

Elementary operations are mostly used to find the inverse of the matrix. The two types of matrix elementary operations are:

• Elementary Row Operations: Elementary operations performed on the rows of the array or matrix are known as primary or elementary row operations.

• Elementary Column Operations: The elementary matrix operations performed on its columns are known as primary or elementary column operations. 

Elementary Operation of Matrix Rules

The following are the rules of the elementary operations of the matrix.

• Any two columns or rows in a matrix or array can be interchanged or exchanged. When we interchange ith row with jth row, then it is written as Ri ↔ Rj. The exchanging of the ith column with the jth column can be written as Ci ↔ Cj.

For example, below is the matrix A

A = [begin{bmatrix} 1 & 2\ 5 & 3 end{bmatrix}]

By applying the elementary matrix operations R1 ↔ R2, we get

A = [begin{bmatrix} 5 & 3\ 1 & 2 end{bmatrix}]

We can multiply the elements of any row (or column) by any non-zero number. We can write the multiplication of ith row with k (any non-zero number) as Ri ↔ k Ri. If we multiply the jth column with k, we can denote it symbolically as Cj ↔ k Cj. 

For example, we have given a matrix A

A = [begin{bmatrix} 2 & 5\ 6 & 3 end{bmatrix}]

If we apply the elementary operation R1 ↔ 3 R1, then we get

A = [begin{bmatrix} 6 & 15\ 6 & 3 end{bmatrix}]

• We can add the elements of any row (or column) with the corresponding elements of another row (or column) of the matrix after multiplying it with any non-zero number. The addition of the elements of an ith row with the jth row, which is multiplied by k (any non-zero number), can be symbolically denoted as Ri ↔ Ri + k Rj. Similarly, we can add the elements of the ith column to the jth column, which is multiplied by k that we can symbolically write as Ci ↔ Ci + k Cj.

For example, we have given a matrix A

A = [begin{bmatrix} 2 & 3\ 6 & 2 end{bmatrix}]

By applying the elementary operation R2 ↔ R2 + 2R1, we get

A = [begin{bmatrix} 4 & 3\ 14 & 8 end{bmatrix}]

Solved Examples

In this section of this article, we have given some matrix elementary operations examples that help you to understand the topic more clearly. 

Example 1: Apply the elementary operation C2 ↔ C1 on a 3 X 3 matrix A. Given that .

A = [begin{bmatrix} 3 & 7 & 2\ 4& 8 & 3\ 6& 9 & 1 end{bmatrix}]

Answer: We have given that

A = [begin{bmatrix} 3 & 7 & 2\ 4& 8 & 3\ 6& 9 & 1 end{bmatrix}]

Now, we have to apply the elementary matrix operation C2 ↔ C1. It means we have to interchange the column 2 with column 1. After using this column operation C2 ↔ C1 on A, we get

A = [begin{bmatrix} 4 & 8 & 3\ 3& 7 & 2\ 6& 9 & 1 end{bmatrix}]

Example 2: Apply the elementary operation R2 ↔ 1/2R2 on matrix A. Given that 

A = [begin{bmatrix} 2 & 3 & 8\ 6& 2 & 10\ 9& 6 & 5 end{bmatrix}].

Answer: Given that

A = [begin{bmatrix} 2 & 3 & 8\ 6& 2 & 10\ 9& 6 & 5 end{bmatrix}]

Now, we have to apply the elementary operation R2 ↔ 1/2R2 on A. It means we have to multiply ½ with every element present in the second row of A, i.e., A21 ↔ ½ A21, A22 ↔ ½ A22, A23 ↔ ½ A23

Hence, A21 will become ½ X 6= three after applying the given elementary operation. Similarly, A22 will become 1, and A23 will become 5.

The matrix obtained after applying the given elementary operation is.   

A = [begin{bmatrix} 2 & 3 & 8\ 3& 1 & 5\ 9& 6 & 5 end{bmatrix}]

Example 3:  Find the matrix obtained after applying the elementary operation C2 ↔ C2 + 2C1 on the below array or matrix..

A = [begin{bmatrix} 3 & 1 & 6\ 4& 9 & 5\ 2& 3 & 4 end{bmatrix}].

Answer: We have given that 

A = [begin{bmatrix} 3 & 1 & 6\ 4& 9 & 5\ 2& 3 & 4 end{bmatrix}]

Now, we have to apply the elementary operation of matrix C2 ↔ C2 + 2C1 to A. It means that every second column element will become the addition of its given elements with corresponding elements of the first column after multiplying with 2. Hence, A12 ↔ A12 + 2A11, A22 ↔ A22 + 2A21and A32 ↔ A32 + 2A31

Therefore, A12 will become 1 + 2 X 3= 7

Similarly, A22 will become 9 + 2 X 4= 17 and A32 will become 3 + 2 X 2= 7

The final matrix obtained after applying the given elementary operation is.

A = [begin{bmatrix} 3 & 7 & 6\ 4& 17 & 5\ 2& 7 & 4 end{bmatrix}]

Different Forms of Equation of a Straight Line

There are different forms of an equation of a line. Some of them are explained below: –

1. Equation For Horizontal Line: A horizontal line is parallel the x-axis. Hence, all the points on this line need to be equidistant from the x-axis. The ordinate defines the distance of a point from the x-axis. Since all the points of the horizontal line are equidistant from the x-axis, the ordinates for all of them will be the same, i.e. y=k for all the points in a particular horizontal line.

2. Equation For Vertical Line: A vertical line is parallel the y-axis. Hence, all the points on this line need to be equidistant from the y-axis. The abscissa defines the distance of a point from the y-axis. Since all the points of the vertical line are equidistant from the y-axis, the abscissa for all of them will be the same i.e. x=k for all the points in a particular vertical line.

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3. Point – Slope Equation: The point-slope form of equation can be used for a non-horizontal, non-vertical line, when we know the values for the slope, and a point on the line. Let us consider I(x, y) as an imaginary point on this line, and P(x1, y1) as the defined point on the line. Then the point-slope equation can be represented as: 

m=[frac{(y − y_{1})}{(x − x_{1})}]

[m(x – x_{1})] = [(y – y_{1})]

By plugging in the values of the slope and the defined point, you will get an equation in terms of x and y, which is the equation of the line. 

1. Equation of Line in Two Point Form: In the two point form of equation, we take a line and consider an imaginary point I (x, y). Now, take two defined points P (x1, y1) and Q (x2, y2) , which are collinear to point I. Since these points are collinear, the slope of PI= slope of PQ. Putting this in the form of an equation:

[frac{(y – y_{1})}{(x – x_{1})}] = [frac{(y_{2} – y_{1})}{(x_{2} – x_{1})}]

[(y – y_{1})] = [(y_{2} – y_{1}) times frac{(x – x_{1}}{(x_{2} – x_{1}})]

2. Slope Intercept Form: Consider the following image. Line L cuts the y-axis at point P(0,a). Here, a is the distance from the origin at which the line L cuts the y-axis. This distance is referred to as the y-intercept. Here, let us take I(x, y) as some imaginary point on this line and P(0,a) as the defined point. We will plug in values in the point-slope equation:

[m(x – x_{1}) = (y – y_{1})]

m(x − 0)= (y − a)

mx – 0 = y – a

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mx – y + a = 0  (or) y = mx + a

Similarly, let us take a line M that cuts the x-axis at Q(b,0). Here, b is the distance from the origin, at which the line M cuts the x-axis. This distance is called the x-intercept. Here, let us take I(x, y) as some imaginary point on this line and Q(b,0) as the defined point. We will plug in values in the point-slope equation:

m (x − x1)= (y − y1)

m (x − b)= (y − 0)

m (x – b) = y

3. Intercept Form: In this case, we are given a line with x-intercept (b,0) and y-intercept (0,a). Consider a random point I (x, y) on the line. Now, we lug in the values in the two-point form equation.

[(y – y_{1})] =[(y_{2} – y_{1})] x [frac{(x – x_{1})}{(x_{2} – x_{1})}]

(y-0) = (a-0) x [frac{(x-b)}{0-b}]  

y= [(frac{-a}{b})] (x-b)

y= [(frac{a}{b})] (b-x)

[(frac{x}{b}) + (frac{y}{a})] = 1

Solved Examples

Example 1: P(3,4) and Q(6, 4) are two collinear points. Represent it in the form of an equation. 

Solution:

The ordinates are equal; hence, this is a horizontal line. The slope of a horizontal line is zero, and it is represented as y = k. Therefore, the equation is y=4.

Example 2: In a particular line, the x-intercept is 3, and the y-intercept is 4. Find the value of the line. 

Solution:

Since we have both the intercepts, we can use the intercept form of equation and substitute values. 

[(frac{x}{b}) + (frac{y}{a})] = 1

[(frac{x}{3}) + (frac{y}{4})] = 1

[frac{(4x + 3)}{12}] = 1

4x + 3y = 12

4x + 3y – 12 = 0

Hence, the equation for this line is 4x + 3y – 12 = 0.