CBSE Class 10 Maths Notes Chapter 3: Pair of Linear Equations in Two Variables

Equation: Assertion that two mathematical expressions with one or more variables are equal is called an equation. 

Equation of a Line: Linear equations are those in which the powers of all the variables involved are equal. A linear equation's degree is always one.

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CBSE Class 10 Maths Chapter 3 Pair of Linear Equations in Two Variables Notes- Free PDF Download

A Linear Equation in Two Variables in Its General Form 

A linear equation in two variables has the generic form ax + by + c = 0, where a and b cannot be zero at the same time.

Students can use the short notes and MCQ questions, as well as the standalone notes.

For easy revision, get the solution pdf for this chapter from the links below:

• Pair of linear Equations in Two Variables- Notes
• Pair of linear Equations in Two Variables-MCQ Questions for Practice
• Pair of linear Equations in Two Variables-Solutions for MCQ Practice

For a word problem, representing linear equations: Using a linear equation to illustrate a word problem 

• Identify unknown quantities and assign variables to them.
• Replacing the unknowns with variables, represent the relationships between quantities in a mathematical

CBSE Class 10 Maths Notes of All Chapters: 

CBSE CLASS 10 MATHS NOTES CHAPTER 1 REAL NUMBERS

CBSE CLASS 10 MATHS NOTES CHAPTER 2 POLYNOMIALS

CBSE CLASS 10 MATHS NOTES CHAPTER 3 PAIR OF LINEAR EQUATIONS IN TWO VARIABLES

CBSE CLASS 10 MATHS NOTES CHAPTER 4 QUADRATIC EQUATION

CBSE CLASS 10 MATHS NOTES CHAPTER 5 AIRTHMETIC PROGRESSION

CBSE CLASS 10 MATHS NOTES CHAPTER 6 TRIANGLE

CBSE CLASS 10 MATHS NOTES CHAPTER 7 CONTROL AND COORDINATION

CBSE Class 10 Maths Notes Chapter 8 Introduction To Trigonometry

CBSE CLASS 10 MATHS NOTES CHAPTER 9 SOME APPLICATIONS OF TRIGONOMETRY

CBSE Class 10 Maths Notes Chapter 10 Circle

CBSE CLASS 10 MATHS NOTES CHAPTER 11 CONSTRUCTION

CBSE CLASS 10 MATHS NOTES CHAPTER 12 AREAS RELATED TO CIRCLES

CBSE CLASS 10 MATHS NOTES CHAPTER 13 SURFACE AREA AND VOLUMES

CBSE CLASS 10 MATHS NOTES CHAPTER 14 STATISTICS

CBSE CLASS 10 MATHS NOTES CHAPTER 15 PROBABILITY

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Solution of Linear Equation in two variables: 

A pair of numbers, one for x and the other for y, that make the two sides of the equation equal is the solution of a linear equation in two variables. 

Solution in Graphics

Graphically representing a pair of linear equations in two variables: 

A pair of straight lines can be used to depict a pair of linear equations in two variables graphically. 

The following is a graphical method for determining the solution to a pair of linear equations: 

• Plot the two equations together (two straight lines)
• Find the location where the lines intersect.
• The solution is the point of intersection. 

Algebraic Solution 

Finding a solution to a pair of Linear Equations that are consistent.

The solution of a pair of linear equations is in the form (x,y), which simultaneously solves both equations. It is possible to get a solution for a consistent pair of linear equations by using

• Method of elimination
• Method of Substitution
• Method of cross-multiplication
• Graphical approach

Finding a solution to a pair of linear equations using the substitution method: 

Method of substitution: 

y – 4x = 1

x + 4y= 38

1. Using one of the equations, express one variable in terms of the other. Y = 4x + 1 in this situation.
2. Substitute this variable (y) in the second equation to obtain a one-variable linear equation,

x + 4 (4x + 1) = 38

x + 16x + 4 = 38

17x + 4 = 38

17x = 38-4

17x= 34

X= 2

17 multiplied by 2 equals 34 

To find the value of a variable, solve the linear equation in that variable.

X = 2

3. Substitute this value for the other variable’s value in one of the equations. 

Y = 4x + 1

Y = 8 + 1

Y = 9

As a result, the solution to the set of linear equations y – 4x = 1 and x + 4y = 38 is (2,9). 

Finding a solution to a pair of linear equations using the elimination approach 

Method of elimination 

Consider the expressions x + 4y = 10 and 2x – y = 2. 

Step 1: By multiplying the coefficients of any variable, you can make them the same.

By multiplying the equations with constants, you can achieve the same result. When we divide the first equation by two, we get 

20 = 2x + 8y 

Step 2: To eliminate one variable, add or subtract the equations, resulting in a single variable equation. 

Subtract the second equation from the first. 

20 = 2x + 8y 

2x – y = 2 – + – ——————– 0(x) + 9y = 18 

Step 3: Solve for one variable and use the result to solve for the other variable in any equation. 

Y = 2 

X = 10 – 4y

X= 10 – 4*2= 10 – 8

X = 2

The answer is (2, 2). 

Cross-multiplication A method for solving a pair of linear equations. 

In the case of the pair of linear equations 

a₁x + b₁y + c₁=0

a₂x + b₂y + c₂=0,
x and y can be calculated as

x = (b₁c₂−b₂c₁)/(a₁b₂−a₂b₁)

y = (c₁a₂−c₂a₁)/(a₁b₂−a₂b₁) 

Using Linear Equations to Solve Problems: 

Equations that can be reduced to a pair

Questions in two variables that can be reduced to a pair of Linear Equations.

Some equations can be simplified to a linear equation by substituting one variable for another. 

2/x+3/y=4 

5/x−4/y=9 

In this situation, we can make the change. 

1/x is equal to u, and 1/y is equal to v. 

2u + 3v = 4 5u – 4v = 9 is the result of the pair of equations. 

It is possible to solve the above pair of equations. Back replace the values of x and y after you’ve solved the problem.