Chapter 3 | PAIR OF LINEAR EQUATIONS IN TWO VARIABLES | CBSE | Class 10 board exam | 10th class mathematics |

Chapter 3 : PAIR OF LINEAR EQUATIONS IN TWO VARIABLES

What is Linear Equation?

A linear equation is an algebraic equation in which the highest power of the variable(s) is 1, and its graph always forms a straight line.
Example : 3x + 4 = 0 , 3x + 4y = 6

Linear Equation in Two variable:

A linear equation in two variables is an equation of the form ax+by+c=0, where a,b,c are real numbers and x,y are variables. Its graph is always a straight line in the coordinate plane.
Example : a) 2x + 3y = 6 , b) 4x + 5y = 8.

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Solution of pair of linear equation in two variable:

A pair of linear equations in two variables can be represented, and solved, by the:
(i) graphical method
(ii) algebraic method
a) Graphical method
1. General Form A pair of linear equations in two variables can be written as:
a₁x + b₁y + c₁ = 0
a₂x + b₂y + c₂ = 0
Here, x and y are variables, and (a₁, b₁, c₁, a₂, b₂, c₂) are constants.
2.Steps in the Graphical Method
Step 1: Rewrite equations in slope-intercept form
Convert each equation into the form (y = mx + c).
Step 2: Plot the equations
For each equation, find at least two points (by substituting values of (x) and plot them on the coordinate plane.
Draw straight lines through these points.
Step 3: Identify intersection
The solution of the pair of equations is the point(s) where the two lines intersect. 3.Types of Solutions
a) Unique Solution (Intersecting lines)
If the lines intersect at one point, that point ((x, y)) is the unique solution.
Example:
x + y = 5, x - y = 1

These intersect at ((3, 2)).
b) No Solution (Parallel lines)
If the lines are parallel, they never meet , it means no solution.
Example:
2x + y = 4, 2x + y = 6

c) Infinitely Many Solutions (Coincident lines)
If the two equations represent the same line, every point on the line is a solution.
Example:
x + y = 2, 2x + 2y = 4

4. Visual Summary
Intersecting lines → One solution ( consistent pair of equations )
Parallel lines → No solution ( inconsistent pair of equations )
Coincident lines → Infinite solutions ( consistent pair of equations )

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Algebraic Methods:

We'll discuss the following methods for finding the solution(s) of a pair of linear equations :
(i) Substitution Method
(ii) Elimination Method

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Substitution Method

We shall explain the method of substitution by taking examples.
Example 4 : Solve the following pair of equations by substitution method:
7x – 15y = 2        ....(1)
x + 2y = 3 .... (2)
Solution :
Step 1 : We pick either of the equations and write one variable in terms of the other.
Let us consider the Equation (2) :
x + 2y = 3
and write it as
x = 3 – 2y       .....(3)
Step 2 : Substitute the value of x in Equation (1). We get
7(3 – 2y) – 15y = 2
21 – 14y – 15y = 2
-29y = -19
Therefore, y =–19/(-29)
y = 19/29
Step 3 : Substituting this value of y in Equation (3), we get
x = 3 – 2(19/29) = 49/29
Therefore, the solution is x = 49/29 , y = 19/29

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Elimination Method

let us consider another method of eliminating (i.e., removing) one variable. This is sometimes more convenient than the substitution method
Let us take an example:
9x – 4y = 2000       ......(1)
7x – 3y = 2000       ......(2)
Step 1 : Multiply Equation (1) by 3 and Equation (2) by 4 to make the coefficients of y equal. Then we get the equations:
27x – 12y = 6000       ......(3)
28x – 12y = 8000       ......(4)
Step 2 : Subtract Equation (3) from Equation (4) to eliminate y, because the coefficients of y are the same. So, we get
(28x – 27x) – (12y – 12y) = 8000 – 6000
i.e., x = 2000
Step 3 : Substituting this value of x in (1), we get
9(2000) – 4y = 2000
i.e., y = 4000
So, the solution of the equations is x = 2000, y = 4000

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Relationship between cofficients and solutions of linear equation