Study Notes About Linear Algebra PDF

Study Notes About Linear algebra Introduction Introduction to Linear Algebra Linear algebra is a fundamental branch of mathematics that deals with vector spaces and linear transformations. It serves as the found...

Study Notes About Linear algebra Introduction Introduction to Linear Algebra Linear algebra is a fundamental branch of mathematics that deals with vector spaces and linear transformations. It serves as the foundation for numerous fields such as physics, engineering, computer science, economics, and data science. More than just the study of linear equations and their solutions, linear algebra explores the properties of lines, planes, vector spaces, mappings, and linear functions, all of which are essential for linear transformations. This subject plays a pivotal role in nearly every area of mathematics, and its applications extend to various disciplines. In physics and engineering, linear algebra is used to model basic objects like planes, lines, and the rotations of objects. In computer science, it is crucial for efficient computation. Throughout this course, you will be introduced to the fundamentals of linear algebra, including its components, common problems, linear equations, and real-world applications. At its core, linear algebra focuses on the study of linear combinations, which involve vectors, matrices, and linear functions. It also delves into the study of linear systems of equations and their transformation properties. The applications of linear algebra Linear algebra has wide-reaching applications, particularly in data science. It is used in various domains, such as facial recognition, software testing, search engine ranking algorithms, and computer graphics. Techniques like linear regression, principal component analysis (PCA), and singular value decomposition (SVD) all rely on linear algebra. Furthermore, linear algebra provides a mathematical framework for formulating and solving problems in fields such as business, social sciences, and physical sciences. In machine learning and artificial intelligence, it helps solve complex problems involving data clustering, classification, validation, and fitting, making it an invaluable tool for modern data-driven industries.  Chapter one Basic Sets Theory Sets in mathematics, are simply a collection of distinct objects forming a group. A set can have any group of items, be it a collection of numbers, days of a week, types of vehicles, and so on. Every item in the set is called an element of the set. Curly brackets are used while writing a set. A very simple example of a set would be like this. Set A = {1,2,3,4,5}. There are various notations to represent elements of a set. Sets are usually represented using a roster form or a set builder form. Let us discuss each of these terms in detail. Sets Definition In mathematics, a set is a well-defined collection of objects. Sets are named and represented using a capital letter. The elements that a set comprises can be any kind of thing: people, letters of the alphabet, numbers, shapes, variables, etc. For example (let A is a set): A = {apple, pear, grape} B = {1, 2, 3, 4, 5} C = {1, b, c, d, e, f} D= {(1, 2), (3, 4), (9, 10)} E= {, , } Set = a collection of anything that is meaningful. Sets in Maths Examples We know that a collection of even natural numbers less than 10 is defined, whereas collection of intelligent students in a class is not defined. Thus, collection of even natural numbers less than 10 can be represented in the form of a set, A = {2, 4, 6, 8}. Let us use this example to understand the basic terminology associated with sets in math. Sets in Maths Examples Some standard sets in maths are:  Set of natural numbers, ℕ = {1, 2, 3,...}  Set of whole numbers, W = {0, 1, 2, 3,...}  Set of integers, ℤ = {..., -3, -2, -1, 0, 1, 2, 3,...}  Set of rational numbers, ℚ = {p/q | q is an integer and q ≠ 0}  Set of irrational numbers, ℚ' = {x | x is not rational}  Set of real numbers, ℝ = ℚ ∪ ℚ' All these are infinite sets. But there can be finite sets as well. For example, the collection of even natural numbers less than 10 can be represented in the form of a set, A = {2, 4, 6, 8}, which is a finite set.  Let us use this example to understand the basic terminology associated with sets in math. Elements of a Set The items present in a set are called either elements or members of a set. The elements of a set are enclosed in curly brackets separated by commas. To denote that an element is contained in a set, the symbol '∈' is used. In the above example, 2 ∈ A. If an element is not a member of a set, then it is denoted using the symbol '∉'. Here, 3 ∉ A. The objects that make up a set are called members or elements of the set. Two sets are equal if they have the same members. That is, a set is completely determined by its members. Order and repetition do not matter in a set. Representation of Sets There are different set notations used for the representation of sets. They differ in the way in which the elements are listed. The three set notations used for representing sets are:  Semantic form  Roster form  Set builder form Semantic Form The semantic notation describes a statement to show what the elements of a set are. For example, Set A is the list of the first five odd numbers. Roster Form The most common form used to represent sets is the roster notation in which the elements of the sets are enclosed in curly brackets separated by commas. For example, Set B = {2,4,6,8,10}, which is the collection of the first five even numbers. In a roster form, the order of the elements of the set does not matter, for example, the set of the first five even numbers can also be defined as {2,6,8,10,4}. Also, if there is an endless list of elements in a set, then they are defined using a series of dots at the end of the last element. For example, infinite sets are represented as, X = {1, 2, 3, 4, 5...}, where X is the set of natural numbers. To sum up the notation of the roster form, please take a look at the examples below. Finite Roster Notation of Sets : Set A = {1, 2, 3, 4, 5} (The first five natural numbers) Infinite Roster Notation of Sets : Set B = {5, 10, 15, 20....} (The multiples of 5) Set Builder Form The set builder notation has a certain rule or a statement that specifically describes the common feature of all the elements of a set. The set builder form uses a vertical bar in its representation, with a text describing the character of the elements of the set. For example, A = { k | k is an even number, k ≤ 20}. The statement says, all the elements of set A are even numbers that are less than or equal to 20. Sometimes a ":" is used in the place of the "|".  Notation – S = {x  T | P(x)} (or S = {x | x  T and P(x)}) – The members of S are members of an already known set T that satisfy property P. An example: – Let Z be the set of integers Let Z+ be the set of positive integers. – Z+ = {x  Z | x > 0} Z = The set of all integers Z = {…, -2, -1, 0, 1, 2, …} Z+ = The set of positive integers Z+ = {1, 2, 3…} = {x | x  Z and x > 0} = {x  Z | x > 0} Z- = The set of negative integers Z- = {…, -3, -2, -1} = {-1, -2, -3…} = {x  Z | x < 0} R = The set of all real numbers Q = the set of all rational numbers Q = {x  R | x = p/q and p, q  Z and q  0} We can use “;” to replace “and” Representation of Sets Using Venn Diagram Venn Diagram is a pictorial representation of sets, with each set represented as a circle. The elements of a set are present inside the circles. Sometimes a rectangle encloses the circles, which represents the universal set. The Venn diagram represents how the given sets are related to each other. Sets Symbols Set symbols are used to define the elements of a given set. The following table shows some of these symbols and their meaning. Symbols Meaning U Universal set n(X) Cardinal number of set X b∈A 'b' is an element of set A a∉B 'a' is not an element of set B {} Denotes a set ∅ Null or empty set AUB Set A union set B A∩B Set A intersection set B A⊆B Set A is a subset of set B B⊇A Set B is the superset of set A Types of Sets Sets are classified into different types. Some of these are singleton, finite, infinite, empty, etc. Singleton Sets A set that has only one element is called a singleton set or also called a unit set. Example, Set A = { k | k is an integer between 3 and 5} which is A = {4}. Finite Sets As the name implies, a set with a finite or countable number of elements is called a finite set. Example, Set B = {k | k is a prime number less than 20}, which is B = {2,3,5,7,11,13,17,19} Infinite Sets  A set with an infinite number of elements is called an infinite set. Example: Set C = {Multiples of 3}. Empty or Null Sets A set that does not contain any element is called an empty set or a null set. An empty set is denoted using the symbol '∅'. It is read as 'phi'. Example: Set X = {}. Equal Sets If two sets have the same elements in them, then they are called equal sets. Example: A = {1,2,3} and B = {1,2,3}. Here, set A and set B are equal sets. This can be represented as A = B. Unequal Sets If two sets have at least one element that is different, then they are unequal sets.Example: A = {1,2,3} and B = {2,3,4}. Here, set A and set B are unequal sets. This can be represented as A ≠ B. Equivalent Sets Two sets are said to be equivalent sets when they have the same number of elements, though the elements are different. Example: A = {1,2,3,4} and B = {a,b,c,d}. Here, set A and set B are equivalent sets since n(A) = n(B) Overlapping Sets Two sets are said to be overlapping if at least one element from set A is present in set B. Example: A = {2,4,6} B = {4,8,10}. Here, element 4 is present in set A as well as in set B. Therefore, A and B are overlapping sets. Disjoint Sets Two sets are disjoint sets if there are no common elements in both sets. Example: A = {1,2,3,4} B = {5,6,7,8}. Here, set A and set B are disjoint sets. Subset For two sets A and B, if every element in set A is present in set B, then set A is a subset of set B(A ⊆ B) and B is the superset of set A(B ⊇ A). Example: A = {1,2,3} B = {1,2,3,4,5,6} A ⊆ B, since all the elements in set A are present in set B. B ⊇ A denotes that set B is the superset of set A. A is a subset () of B, or B is a superset of A iff every member of A is a member of B. – A  B iff fo rall x if x  A, then x  B An example: – {-2, 0, 6}  {-3, -2, -1, 0, 1, 3, 6} Negation: A is not a subset of B or B is not a superset of A iff there is a member of A that is not a member of B – A  B iff there exist x, x  A, x  B  Universal Set A universal set is the collection of all the elements in regard to a particular subject. The universal set is denoted by the letter 'U'. Example: Let U = {The list of all road transport vehicles}. Here, a set of cars is a subset for this universal set, the set of cycles, trains are all subsets of this universal set. Depending on the context of discussion – Define a set of U such that all sets of interest are subsets of U. – The set U is known as a universal set Examples: – When dealing with integers, U may be Z. – When dealing with plane geometry, U may be the set of points in the plane Power Sets Power set is the set of all subsets that a set could contain. Example: Set A = {1,2,3}. Power set of A is = {{∅}, {1}, {2}, {3}, {1,2}, {2,3}, {1,3}, {1,2,3}}. The set of all subsets of a set is called the power set of the set The power set of S is denoted by P(S). Example: – P() ={} – P({1, 2}) = {, {1}, {2}, {1, 2}} – P(S) = {, …, S} – What is P({1, 2, 3})? How many elements does the power set of S have? Assume S has n elements. Sets Formulas Sets find their application in the field of algebra, statistics, and probability. There are some important set formulas as listed below. For any two overlapping sets A and B,  n(A U B) = n(A) + n(B) - n(A ∩ B)  n (A ∩ B) = n(A) + n(B) - n(A U B)  n(A) = n(A U B) + n(A ∩ B) - n(B)  n(B) = n(A U B) + n(A ∩ B) - n(A)  n(A - B) = n(A U B) - n(B)  n(A - B) = n(A) - n(A ∩ B) For any two sets A and B that are disjoint,  n(A U B) = n(A) + n(B)  A∩B=∅  n(A - B) = n(A) Properties of Sets Similar to numbers, sets also have properties like associative property, commutative property, and so on. There are six important properties of sets. Given, three sets A, B, and C, the properties for these sets are as follows. Property Example AUB=BUA Commutative Property A∩B=B∩A (A ∩ B) ∩ C = A ∩ (B ∩ C) Associative Property (A U B) U C = A U (B U C) A U (B ∩ C) = (A U B) ∩ (A U C) Distributive Property A ∩ (B U C) = (A ∩ B) U (A ∩ C) AU∅=A Identity Property A∩U=A Complement Property A U A' = U A∩A=A Idempotent Property AUA=A Operations on Sets Some important operations on sets include union, intersection, difference, the complement of a set, and the cartesian product of a set. A brief explanation of operations on sets is as follows. Union of Sets Union of sets, which is denoted as A U B, lists the elements in set A and set B or the elements in both set A and set B. For example, {1, 3} ∪ {1, 4} = {1, 3, 4} Intersection of Sets The intersection of sets which is denoted by A ∩ B lists the elements that are common to both set A and set B. For example, {1, 2} ∩ {2, 4} = {2} Set Difference Set difference which is denoted by A - B, lists the elements in set A that are not present in set B. For example, A = {2, 3, 4} and B = {4, 5, 6}. A - B = {2, 3}. Set Complement Set complement which is denoted by A', is the set of all elements in the universal set that are not present in set A. In other words, A' is denoted as U - A, which is the difference in the elements of the universal set and set A.  Examples on Sets  Example:  Find the elements of the sets represented as follows and write the cardinal number of each set.  a) Set A is the first 8 multiples of 7  b) Set B = {a,e,i,o,u}  c) Set C = {x | x are even numbers between 20 and 40} Solution: a) Set A = {7,14,21,28,35,42,49,56}. These are the first 8 multiples of 7. Since there are 8 elements in the set, cardinal number n (A) = 8 b) Set B = {a,e,i,o,u}. There are five elements in the set,  Therefore, the cardinal number of set B, n(B) = 5. c) Set C = {22,24,26,28,30,32,34,36,38}. These are the even numbers between 20 and 40, which make up the elements of the set C. Therefore, the cardinal number of set C, n(C) = 9.  Example:  If Set A = {a,b,c}, Set B = {a,b,c,p,q,r}, U = {a,b,c,d,p,q,r,s}, find the following using sets formulas,  a) A U B  b) A ∩ B  c) A'  d) Is A ⊆ B?  (Here 'U' is the universal set). Solution: a) A U B = {a,b,c,p,q,r} b) A ∩ B = {a,b,c} c) A' = {d,p,q,r,s} d) A ⊆ B, (Set A is a subset of set B) since all the elements in set A are present in set B.  Example:  Express the given set in set-builder form: A = {2, 4, 6, 8, 10, 12, 14} Solution: Given: A = {2, 4, 6, 8, 10, 12, 14} Using sets notations, we can represent the given set A in set-builder form as, A = {x | x is an even natural number less than 15} Example: Example: Example: There are 30 students in a class. Among them, 8 students are learning both English and French. A total of 18 students are learning English. If every student is learning at least one language, how many students are learning French in total? Solution: The Venn diagram for this problem looks like this. Every student is learning at least one language. Hence there is no one who fall in the category ‘neither’. So in this case, n(EᴜF) = n(µ). It is mentioned in the problem that a total of 18 are learning English. This DOES NOT mean that 18 are learning ONLY English. Only when the word ‘only’ is mentioned in the problem should we consider it so. Now, 18 are learning English and 8 are learning both. This means that 18 – 8 = 10 are learning ONLY English. n(µ) = 30, n(E) = 10 n(EᴜF) = n(E) + n(F) – n(E∩F) 30 = 18+ n(F) – 8 n(F) = 20 Therefore, total number of students learning French = 20. Note: The question was only about the total number of students learning French and not about those learning ONLY French, which would have been a different answer, 12. Finally, the Venn diagram looks like this. Example: Let S={1,2,3} Write all the possible partitions of S. Solution Remember that a partition of S is a collection of nonempty sets that are disjoint and their union is S. There are 55 possible partitions for S={1,2,3}={1,2,3}: {1},{2},{3}{1},{2},{3}; {1,2},{3}{1,2},{3}; {1,3},{2}{1,3},{2}; {2,3},{1}{2,3},{1}; {1,2,3}{1,2,3}. Example: Write the elements of the following sets: To write down the elements of S, note that W represents the set of positive integers and zero. Therefore, we get: Now, we write the elements of F: In order to write down the elements of Q, we use the definition for d and evaluate the elements: The set K, is defined in terms of the set of natural numbers N. The set of natural numbers is the set of positive integers greater than and equal to one. Thus, we have the following elements in K:  Example: Consider the sets given in the previous example, find the following sets: We get the following sets: Questions with answer: Multiple-Choice Questions (MCQs) 1. Which of the following is a finite set? o a) {1, 2, 3, …} o b) {x | x is an odd number less than 10} o c) {1, 2, 3, …, n} o d) ℕ = {1, 2, 3,...} o Answer: b) {x | x is an odd number less than 10} 2. What does the symbol '∅' represent in set theory? o a) Null set o b) Universal set o c) Finite set o d) Infinite set o Answer: a) Null set 3. If A = {2, 4, 6} and B = {4, 6, 8}, what is A ∩ B? o a) {2, 4, 6, 8} o b) {4, 6} o c) {2, 8} o d) ∅ o Answer: b) {4, 6} 4. Which of the following is an example of an infinite set?  o a) {x | x is a prime number less than 10} o b) ℕ = {1, 2, 3,...} o c) {a, b, c, d} o d) {1, 2, 3, 4} o Answer: b) ℕ = {1, 2, 3,...} 5. The set of all positive integers is denoted as: o a) ℕ o b) ℤ o c) ℝ o d) ℚ o Answer: a) ℕ 6. What is the correct representation of a power set for A = {1, 2}? o a) {{1}, {2}} o b) {{∅}, {1}, {2}, {1, 2}} o c) {{1, 2, 3}} o d) {1, 2, 3} o Answer: b) {{∅}, {1}, {2}, {1, 2}} 7. Which of the following statements is true about the universal set? o a) A universal set contains all elements of interest. o b) A universal set is a null set. o c) A universal set has only finite elements. o d) A universal set is the complement of a set. o Answer: a) A universal set contains all elements of interest. 8. If A = {1, 2, 3} and B = {4, 5}, what is A ∪ B? o a) {1, 2, 3} o b) {4, 5} o c) {1, 2, 3, 4, 5} o d) ∅ o Answer: c) {1, 2, 3, 4, 5} 9. Which of the following is a subset of ℤ (set of integers)? o a) {x | x is a natural number} o b) {x | x is an irrational number} o c) {x | x is a real number} o d) {x | x is a complex number} o Answer: a) {x | x is a natural number} 10. In set-builder notation, A = {x | x is a positive even integer ≤ 10}. What is A? o a) {1, 3, 5, 7, 9} o b) {2, 4, 6, 8, 10} o c) {2, 4, 6, 8} o d) {2, 4, 6} o Answer: b) {2, 4, 6, 8, 10} True/False Questions 1. The set ℕ represents the set of all integers. o False (ℕ represents the set of natural numbers, not integers.) 2. A power set includes all possible subsets of a given set.  o True 3. The union of two disjoint sets is always a null set. o False (The union of two disjoint sets is a set containing all elements from both sets.) 4. In a Venn diagram, the universal set is represented by a rectangle. o True 5. A singleton set contains only one element. o True 6. The set of real numbers (ℝ) is a subset of rational numbers (ℚ). o False (Rational numbers are a subset of real numbers, but not all real numbers are rational.) 7. Two sets are considered equal if they have the same number of elements, regardless of their values. o False (Sets are equal if they have the same elements, not just the same number of elements.) 8. The intersection of a set with itself is the set itself. o True 9. The complement of a universal set is an empty set. o True 10. {2, 3, 4} ⊆ {2, 4, 6, 8} is a true statement. o False (Set {2, 3, 4} is not a subset of {2, 4, 6, 8} because 3 is not in the second set.) Problems with Answers 1. Problem: If A = {1, 3, 5, 7} and B = {3, 7, 9}, find A ∩ B. o Solution: A ∩ B = {3, 7} 2. Problem: If C = {x | x is an even number ≤ 10}, list the elements of set C. o Solution: C = {2, 4, 6, 8, 10} 3. Problem: What is the power set of A = {1, 2}? o Solution: P(A) = {{∅}, {1}, {2}, {1, 2}} 4. Problem: Find A U B if A = {2, 4, 6} and B = {4, 5, 6}. o Solution: A U B = {2, 4, 5, 6} 5. Problem: If D = {1, 2, 3, 4, 5}, what is D' (complement of D) in the universal set U = {1, 2, 3, 4, 5, 6, 7, 8}? o Solution: D' = {6, 7, 8} 6. Problem: If n(A) = 7, n(B) = 5, and n(A ∩ B) = 2, find n(A U B). o Solution: n(A U B) = n(A) + n(B) - n(A ∩ B) = 7 + 5 - 2 = 10 7. Problem: List the subsets of A = {a, b}. o Solution: Subsets of A = {∅, {a}, {b}, {a, b}} 8. Problem: If A = {1, 2, 3} and B = {3, 4, 5}, find A - B. o Solution: A - B = {1, 2} 9. Problem: Find n(A U B) if A = {2, 4, 6, 8} and B = {4, 6, 8, 10}. o Solution: A U B = {2, 4, 6, 8, 10}, n(A U B) = 5 10. Problem: If E = {1, 2, 3, 4}, express E in set-builder form. o Solution: E = {x | x is a natural number less than 5}  Chapter two Introduction to Liner equations Introduction to Liner equations A linear equation is an equation that describes a straight line on a graph. You can remember this by the "line" part of the name linear equation. Where A and B are coefficients (numbers) while x and y are variables. A linear equation is an equation in which the highest power of the variable is always 1. It is also known as a one-degree equation. The standard form of a linear equation in one variable is of the form Ax + B = 0. Here, x is a variable, A is a coefficient and B is constant. The standard form of a linear equation in two variables is of the form Ax + By = C. Here, x and y are variables, A and B are coefficients and C is a constant. An equation that has the highest degree of 1 is known as a linear equation. This means that no variable in a linear equation has an exponent more than 1. The graph of a linear equation always forms a straight line. Linear Equation Definition: A linear equation is an algebraic equation where each term has an exponent of 1 and when this equation is graphed, it always results in a straight line. This is the reason why it is named as a 'linear equation'. There are linear equations in one variable and linear equations in two variables. Let us learn how to identify linear equations and non-linear equations with the help of the following examples. Equations Linear or Non-Linear y = 8x - 9 Linear  Equations Linear or Non-Linear Non-Linear, the power of the variable x y = x2 - 7 is 2 Non-Linear, the power of the variable y √y + x = 6 is 1/2 y + 3x - 1 = 0 Linear Non-Linear, the power of the variable y y2 - x = 9 is 2 Linear Equation Formula The linear equation formula is the way of expressing a linear equation. This can be done in different ways. For example, a linear equation can be expressed in the standard form, the slope-intercept form, or the point-slope form. Now, if we take the standard form of a linear equation, let us learn the way in which it is expressed. We can see that it varies from case to case based on the number of variables and it should be remembered that the highest (and the only) degree of all variables in the equation should be 1. Linear Equations in Standard Form The standard form or the general form of linear equations in one variable is written as, Ax + B = 0; where A and B are real numbers, and x is the single variable. The standard form of linear equations in two variables is expressed as, Ax + By = C; where A, B and C are any real numbers, and x and y are the variables. Linear Equation Graph The graph of a linear equation in one variable x forms a vertical line that is parallel to the y-axis and vice-versa, whereas, the graph of a linear equation in two variables x and y forms a straight line. Let us graph a linear equation in two variables with the help of the following example. Example: Plot a graph for a linear equation in two variables, x - 2y = 2. Let us plot the linear equation graph using the following steps. Step 1: The given linear equation is x - 2y = 2. Step 2: Convert the equation in the form of y = mx + b. This will give: y = x/2 - 1. Step 3: Now, we can replace the value of x for different numbers and get the resulting value of y to create the coordinates. Step 4: When we put x = 0 in the equation, we get y = 0/2 - 1, i.e. y = -1. Similarly, if we substitute the value of x as 2 in the equation, y = x/2 - 1, we get y = 0. Step 5: If we substitute the value of x as 4, we get y = 1. The value of x = -2 gives the value of y = -2. Now, these pairs of values of (x, y) satisfy the given linear equation y = x/2 - 1. Therefore, we list the coordinates as shown in the following table. Step 6: Finally, we plot these points (4,1), (2,0), (0,-1) and (-2, -2) on a graph and join the points to get a straight line. This is how a linear equation is represented on a graph. Linear Equations in One Variable A linear equation in one variable is an equation in which there is only one variable present. It is of the form Ax + B = 0, where A and B are any two real numbers and x is an unknown variable that has only one solution. It is the easiest way to represent a mathematical statement. This equation has a degree that is always equal to 1. A linear equation in one variable can be solved very easily. The variables are separated and brought to one side of the equation and the constants are combined and brought to the other side of the equation, to get the value of the unknown variable. Example: Solve the linear equation in one variable: 3x + 6 = 18. In order to solve the given equation, we bring the numbers on the right-hand side of the equation and we keep the variable on the left-hand side. This means, 3x = 18 - 6. Then, as we solve for x, we get, 3x = 12. Finally, the value of x = 12/3 = 4. Linear Equations in Two Variables A linear equation in two variables is of the form Ax + By + C = 0, in which A, B, C are real numbers and x and y are the two variables, each with a degree of 1. If we consider two such linear equations, they are called simultaneous linear equations. For example, 6x + 2y + 9 = 0 is a linear equation in two variables. There are various ways of solving linear equations in two variables like the graphical method, the substitution method, the cross multiplication method, the elimination method, and the determinant method. How to Solve Linear Equations? An equation is like a weighing balance with equal weights on both sides. If we add or subtract the same number from both sides of an equation, it still holds true. Similarly, if we multiply or divide the same number on both sides of an equation, it is correct. We bring the variables to one side of the equation and the constant to the other side and then find the value of the unknown variable. This is the way to solve a linear equation with one variable. Let us understand this with the help of an example. Example: Solve the equation, 3x - 2 = 4. We perform mathematical operations on the Left-hand side (LHS) and the right- hand side (RHS) so that the balance is not disturbed. So, let us add 2 on both sides to reduce the LHS to 3x. This will not disturb the balance. The new LHS is 3x - 2 + 2 = 3x and the new RHS is 4 + 2 = 6. Now, let us divide both sides by 3 to reduce the LHS to x. Thus, we have x = 2. This is one of the ways of solving linear equations in one variable. The value of the variable that makes a linear equation true is called the solution or root of the linear equation. The solution of a linear equation is unaffected if the same number is added, subtracted, multiplied, or divided into both sides of the equation. The graph of a linear equation in one or two variables always forms a straight line. Linear Equation Examples Example : The sum of two numbers is 44. If one number is 10 more than the other, find the numbers by framing a linear equation. Solution: Let the number be x, so the other number is x + 10. We know that the sum of both the numbers is 44. Therefore, the linear equation can be framed as, x + x + 10 = 44. This results in, 2x + 10 = 44. Now, let us solve the equation by isolating the variable on one side and by bringing the constants on the other side. This means 2x = 44 - 10. By simplifying RHS, we get, 2x = 34, so the value of x is 17. This means, one number is 17 and the other number is 17 + 10 = 27. Answer: Therefore, the two numbers are 17 and 27. Example: Six times of a number is equal to 48. Form a linear equation and find the unknown number. Solution: Let the unknown number be x. Six times of this number is equal to 48 means 6x = 48. So, this linear equation can be solved to find the value of x which is the unknown number. 6x = 48 means x = 48/6 = 8. Answer: Therefore, the unknown number is 8. Example: Solve the given linear equation: 5x - 95 = 75. Solution: The given equation is 5x - 95 = 75. ⇒ 5x = 75 + 95 ⇒ 5x = 170 ⇒ x = 34 Answer: Therefore, the value of x is 34. Solved examples Example: Solve x – y = 12 and 2x + y = 22 Solution: Name the equations x – y = 12 …(1) 2x + y = 22 …(2) Isolate Equation (1) for x, x = y + 12 Substitute x =y + 12 in equation (2) 2(y+12) + y = 22 3y + 24 = 22 3y = -2 Or y = -2/3 Substitute the value of y in x = y + 12 x = y + 12 x = -2/3 + 12 x = 34/3 Answer: x = 34/3 and y = -2/3 ----------------------------------------------------------- Problems and Solutions Question: Find the value of variables which satisfies the following equation: 2x + 5y = 20 and 3x+6y =12. Solution: Using the method of substitution to solve the pair of linear equation, we have: 2x + 5y = 20…………………….(i) 3x+6y =12……………………..(ii) Multiplying equation (i) by 3 and (ii) by 2, we have: 6x + 15y = 60…………………….(iii) 6x+12y = 24……………………..(iv) Subtracting equation (iv) from (iii) 3y = 36 ⇒ y = 12 Substituting the value of y in any of the equation (i) or (ii), we have 2x + 5(12) = 20 ⇒ x = −20 Therefore, x=-20 and y =12 is the point where the given equations intersect. Now, it is important to know the situational examples which are also known as word problems from linear equations in 2 variables. Pair of Linear Equation in Two Variables Notes In earlier classes, we studied linear equation in one variable, and we learned how to solve them. If there were one variable and one equation, we used to solve it easily, but in this case, we have two variables and two equations. One thing is for certain we need two different sets of linear equations in order to find out the two different unknowns. If one equation is given and two variables are asked to be solved, we will not get the particular solution. For example, 3x + 2y = 9 and 5x + y = 10  These simultaneous equations can be solved, and we can arrive at a particular solution from these but on the other hand, 6x + 7y = 9 Here, we cannot get a particular solution for this as there is only one condition given, and we have two unknowns. We can rewrite the above equation as: y = (9-6x)/7 Depending on the values of x, the values of y will change accordingly. So, one unique solution is not possible. Thus, it can be clearly said that in order to get a particular solution of systems, of linear equations in two variables, we need two different sets of independent conditions. Example: Solve the following system of equations using the substitution method. x+2y-7=0 2x-5y+13=0 Solution: Let us solve the equation, x+2y-7=0 for y: x+2y-7=0 ⇒2y=7-x ⇒ y=(7-x)/2 Substitute this in the equation, 2x-5y+13=0: 2x-5y+13=0 ⇒ 2x-5((7-x)/2)+13=0 ⇒ 2x-(35/2)+(5x/2)+13=0 ⇒ 2x + (5x/2) = 35/2 - 13 ⇒ 9x/2 = 9/2 ⇒ x=1 Substitute x=1 this in the equation y=(7-x)/2: y=(7-1)/2 = 3 Therefore, the solution of the given system is x=1 and y=3. Method of Elimination To solve a system of linear equations in two variables using the elimination method, we will use the steps given below:  Step 1: Arrange the equations in the standard form: ax+by+c=0 or ax+by=c.  Step 2: Check if adding or subtracting the equations would result in the cancellation of a variable.  Step 3: If not, multiply one or both equations by either the coefficient of x or y such that their addition or subtraction would result in the cancellation of any one of the variables.  Step 4: Solve the resulting single variable equation.  Step 5: Substitute it in any of the given equations to get the value of another variable. Example: Solve the following system of equations using the elimination method. 2x+3y-11=0 3x+2y-9=0 Adding or subtracting these two equations would not result in the cancellation of any variable. Let us aim at the cancellation of x. The coefficients of x in both equations are 2 and 3. Their LCM is 6. We will make the coefficients of x in both equations 6 and -6 such that the x terms get cancelled when we add the equations. 3 × (2x+3y-11=0) ⇒ 6x+9y-33=0 -2 × (3x+2y-9=0) ⇒ -6x-4y+18=0 Now we will add these two equations: 6x+9y-33=0 -6x-4y+18=0 On adding both the above equations we get, ⇒ 5y-15=0 ⇒ 5y=15 ⇒ y=3 Substitute this in one of the given two equations and solve the resultant variable for x. 2x+3y-11=0 ⇒ 2x+3(3)-11=0 ⇒ 2x+9-11=0 ⇒ 2x=2 ⇒ x=1 Therefore, the solution of the given system of equations is x=1 and y=3. Problems and Solutions Question: find the price of variables that satisfy the following equation:2x + 5y = 20 and 3x+6y =12.Answer: using the method of substitution to solve the pair of linear equations, we have:2x + 5y = 20…………………….(i)3x+6y =12……………………..(ii)Multiplying equation (i) via 3 and (ii) through 2, we have:6x + 15y = 60…………………….(iii)6x+12y = 24……………………..(iv)Subtracting equation (iv) from (iii)3y = 36⇒ y = 12Substituting the cost of y in any of the equations (i) or (ii), we've got2x + 5(12) = 20⇒ x = −20Consequently, x=-20 and y =12 is the factor wherein the given equations intersect Multiple-Choice Questions (MCQs) 1. Which of the following is a linear equation in one variable? o a) 2x2+5=72x^2 + 5 = 72x2+5=7 o b) 4x+9=24x + 9 = 24x+9=2 o c) y2−4=0y^2 - 4 = 0y2−4=0 o d) x3+2=5x^3 + 2 = 5x3+2=5 o Answer: b) 4x+9=24x + 9 = 24x+9=2 2. What is the degree of a linear equation? o a) 0 o b) 1 o c) 2 o d) 3 o Answer: b) 1 3. Which of the following represents the standard form of a linear equation in two variables? o a) y=mx+by = mx + by=mx+b o b) Ax+By=CAx + By = CAx+By=C o c) Ax+B=0Ax + B = 0Ax+B=0 o d) x2+y=5x^2 + y = 5x2+y=5 o Answer: b) Ax+By=CAx + By = CAx+By=C 4. What does the graph of a linear equation in two variables look like? o a) A curve o b) A straight line o c) A circle o d) A parabola o Answer: b) A straight line 5. If 3x+6=183x + 6 = 183x+6=18, what is the value of xxx? o a) 4 o b) 3 o c) 6  o d) 2 o Answer: b) 4 6. Which of the following is a solution to the system of equations x−y=12x - y = 12x−y=12 and 2x+y=222x + y = 222x+y=22? o a) x=10,y=−2x = 10, y = -2x=10,y=−2 o b) x=34,y=−2x = 34, y = -2x=34,y=−2 o c) x=34/3,y=−2/3x = 34/3, y = -2/3x=34/3,y=−2/3 o d) x=22,y=12x = 22, y = 12x=22,y=12 o Answer: c) x=34/3,y=−2/3x = 34/3, y = -2/3x=34/3,y=−2/3 7. Solve 6x−3=156x - 3 = 156x−3=15. o a) x=3x = 3x=3 o b) x=5x = 5x=5 o c) x=2x = 2x=2 o d) x=1x = 1x=1 o Answer: a) x=3x = 3x=3 8. If the sum of two numbers is 50 and one number is 10 more than the other, what are the two numbers? o a) 20 and 30 o b) 15 and 35 o c) 25 and 25 o d) 10 and 40 o Answer: a) 20 and 30 9. Which method involves solving one variable in terms of another and substituting it into the other equation? o a) Substitution method o b) Elimination method o c) Graphical method o d) Cross multiplication method o Answer: a) Substitution method 10. What is the solution to the system of equations: 2x+3y=112x + 3y = 112x+3y=11 and 3x+2y=93x + 2y = 93x+2y=9? o a) x=1,y=3x = 1, y = 3x=1,y=3 o b) x=2,y=1x = 2, y = 1x=2,y=1 o c) x=1,y=2x = 1, y = 2x=1,y=2 o d) x=0,y=0x = 0, y = 0x=0,y=0 o Answer: c) x=1,y=2x = 1, y = 2x=1,y=2 True/False Questions 1. A linear equation in one variable can have more than one solution. o False (A linear equation in one variable has exactly one solution.) 2. A linear equation in two variables represents a straight line when plotted on a graph. o True 3. y=x2−4y = x^2 - 4y=x2−4 is a linear equation. o False (The variable xxx has a degree of 2, making it non-linear.) 4. The equation 5x−15=05x - 15 = 05x−15=0 has no solution.  o False (It has one solution, x=3x = 3x=3.) 5. In a system of linear equations, it is possible to have no solution. o True (This happens when the lines are parallel and never intersect.) 6. The slope-intercept form of a linear equation is written as Ax+By=CAx + By = CAx+By=C. o False (The slope-intercept form is y=mx+by = mx + by=mx+b.) 7. Adding or subtracting the same number on both sides of a linear equation does not affect its solution. o True 8. y=x+3y = \sqrt{x} + 3y=x+3 is a linear equation. o False (The square root of xxx makes it non-linear.) 9. The graph of x=4x = 4x=4 is a vertical line. o True 10. A linear equation in two variables can have infinitely many solutions. o True (All points on the line are solutions to the equation.) Problems with Answers 1. Problem: Solve 2x+5=15. o Solution: 2x=15−5 o 2x=10 o x = 10 /2 x=5 2. Problem: Find the value of xxx in the equation 4x−7=214x - 7 = 214x−7=21. o Solution: 4x=21+74x = 21 + 74x=21+7 4x=284x = 284x=28 x=28/4=7x = 28/4 = 7x=28/4=7 3. Problem: Solve the system of equations x+y=7x + y = 7x+y=7 and x−y=3x - y = 3x−y=3. o Solution: Adding the equations: (x+y)+(x−y)=7+3(x + y) + (x - y) = 7 + 3(x+y)+(x−y)=7+3 2x=102x = 102x=10 x=5x = 5x=5 Substituting in x+y=7x + y = 7x+y=7: 5+y=75 + y = 75+y=7 y=2y = 2y=2 Answer: x=5,y=2x = 5, y = 2x=5,y=2 4. Problem: Solve 5x+4=9x−85x + 4 = 9x - 85x+4=9x−8. o Solution: 5x−9x=−8−45x - 9x = -8 - 45x−9x=−8−4  −4x=−12-4x = -12−4x=−12 x=3x = 3x=3 5. Problem: The sum of two numbers is 20, and their difference is 4. Find the two numbers. o Solution: Let the numbers be xxx and yyy. x+y=20x + y = 20x+y=20 x−y=4x - y = 4x−y=4 Adding both equations: 2x=242x = 242x=24 x=12x = 12x=12 Substituting in x+y=20x + y = 20x+y=20: 12+y=2012 + y = 2012+y=20 y=8y = 8y=8 Answer: The numbers are 12 and 8. 6. Problem: Solve 7x−3=5x+117x - 3 = 5x + 117x−3=5x+11. o Solution: 7x−5x=11+37x - 5x = 11 + 37x−5x=11+3 2x=142x = 142x=14 x=7x = 7x=7 7. Problem: Solve 3x+4=103x + 4 = 103x+4=10 for xxx. o Solution: 3x=10−43x = 10 - 43x=10−4 3x=63x = 63x=6 x=6/3=2x = 6/3 = 2x=6/3=2 8. Problem: Find the value of xxx and yyy in the system 2x+3y=122x + 3y = 122x+3y=12 and x−y=2x - y = 2x−y=2. o Solution: From x−y=2x - y = 2x−y=2, we get x=y+2x = y + 2x=y+2. Substituting into 2x+3y=122x + 3y = 122x+3y=12: 2(y+2)+3y=122(y + 2) + 3y = 122(y+2)+3y=12 2y+4+3y=122y + 4 + 3y = 122y+4+3y=12 5y=85y = 85y=8 y=8/5y = 8/5y=8/5 Substituting into x=y+2x = y + 2x=y+2: x=8/5+2=18/5x = 8/5 + 2 = 18/5x=8/5+2=18/5 Answer: x=18/5,y=8/5x = 18/5, y = 8/5x=18/5,y=8/5