# CBSE Study Notes For Class 10 Maths Chapter 3 Pair of Linear Equations in Two Variables

General form of a linear equation
ax + by + c = 0.
Two linear equations in the same two variables are called a pair of linear equations in two variables.
The general form of the equations
a1x + b1y + c1 = 0
a2x + b2y + c2 = 0

where a1, a2, b1, b2, c1 and c2 are real numbers
Points to remember

• The graph of a linear equation is a straight line.
• A pair of values of variables ‘x‘ and ‘y’ which satisfy both the equations in the given system of equations is said to be a solution of the pair of linear equations.

Representation of a pair of linear equations in two variables:

(i) Graphical method

(ii) Algebraic method
Methods to solve a pair of linear equations using ALGEBRAIC METHOD-

• Substitution Method
• Elimination Method
• Cross Multiplication Method

GRAPHICAL METHOD (Graphical Representation)

Example: Draw the graph of y=2x+1

Types of Lines

A pair of linear equations in two variables can be represented as:-

• INTERSECTING LINES

They intersect at exactly one point.

PARALLEL LINES
They never intersect each other.

COINCIDENT LINES
They Coincide each other.

Consistency of System
system of two linear equations can have one solution, an infinite number of solutions, or no solution.
If a system has at least one solution, it is said to be consistent .
If a system has no solution, it is said to be inconsistent.
If a consistent system has exactly one solution, it is independent .
If a consistent system has an infinite number of solutions, it is dependent .

• SUBSTITUTION METHOD

In this method, the value of one variable from one equation is substituted in the other equation.
Example: Solve the following system by substitution.
2x – 3y = –2———>1
4x +   y = 24———>2

4x + y = 24
y = –4x + 24——>3
Now substitute the value of y in equation (1) and solve for x:
2x – 3(–4x + 24) = –2
2x + 12x – 72 = –2
14x = 70
x = 5
Now put x=5 in equation (3)
y = –4(5) + 24 = –20 + 24 = 4
So, the solution is (x, y) = (5, 4).

• ELIMINATION METHOD

The elimination method is used to solve linear equations in two variables, where one of the variables is removed or eliminated.
Example:

2x + 7y = 10…………….. (1)
3x + y = 6………………… (2)
First way,
Multiply equation (1) by 3 and equation (2) by 2, we get,
6x + 21y = 30……………..(3)
6x + 2y = 12……………….(4)
The coefficients the x in equation (3) and equation (4) are the same i.e. 6.
Now, subtract equation (4) from equation (3). We get-
6x + 21y – 6x – 2y = 30 – 12
⇒ 19y = 18
y =  18/19
Substitute the value of y in either equation (1) or (2),
Lets put it in equation (2).
3x + 18/19 = 6
3x = 6 – 18/19
3x = 96/19
x = 96/57 = 32/19

Second way,
Multiply equation (2) with 7,
21x + 7y = 42………….(5)
And equation (1) is 2x + 7y = 10
Subtracting equation (1) from equation (5), we get
19x = 32
x = 32/19
Substituting the value of x in Eqn. (1),
2(32/19) + 7y = 10
7y = 10 – 64/19
7y = 126/19
y = 18/19

• CROSS MULTIPLICATION METHOD

This is the simplest method and gives the accurate value of the variables.

Example: Solve the following linear equations using cross multiplication method.
3x − 4y = 2
y − 2x = 7

Solution: The above equation can be rewritten as:
3x − 4y = 2
−2x + y = 7

By method of cross multiplication,

⇒ x = −6 , y = −5
Solving the pair of equations reducible to the pair of linear equations in two variables

• Find the expressions that repeat in both the equations. Give them a simpler form: say x and y.
• Solve the new pair of linear equations for the new variables.

Solve the following system of equations

y=2
Therefore, x=3, y=2

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