Points to Remember: Pair of Linear Equations in Two Variables

 Page 1


A pair of linear 
equation in two 
Variables 
Page 2


A pair of linear 
equation in two 
Variables 
Substitution Method
Linear equation can be represented and solved by
Equation in the form of ax + by + c = 0, where a, b & c are real numbers and ‘a’ & ‘b’ both cannot be
zero is known as linear equation in two variables          
Graphical Method Algebraic Method
Substitution Elimination Cross Multiplication
Cross multiplication method Elimination Method
.   We nd the value of one variable ( say y ) in
 terms of the other variable ( say x ) from either of 
 the equations.
.   Substitute the value of y in another equation
 to make it in one variable. Now we can get the
  value of x.
.   Substitute the value of obtained variable x in 
  any of the equations to get the value of ‘y ’. 
.   We make coecient of a variable in one of the 
 equations equal to the coecient of the same 
 variable in another equation by multiplying it with 
 a suitable number.
.   Add or subtract one equation from other to
 eliminate one variable.
.   Now we get an equation in one variable and  
 can solve it to get its value.
 .   Substitute the value of the obtained variable in
 any of the equations to get the value of another
 variable. 
1
1 1
2
3
2
3
4
.   Let the general from of pair of linear 
equations in two variables x & y be 
a
1
x + b
1
y + c
1
 = 0
a
2
x + b
2
y + c
2
 = 0
2 .   x & y can be found by using following formulas
b
1
c
2
 - b
2
c
1
a
1
b
2
 - a
2
b
1
c
1
a
2
 - c
2
a
1
a
1
b
2
 - a
2
b
1
y =
x =
Methods to solve pair of linear equations
A PAIR OF LINEAR EQUATIONS IN TWO VARIABLES 7
A Pair of Linear Equations in Two Variables
{ Sotution }
Y
X
1
1
(0,0)
2 x + y = 6
x + y = 4 (2, 2)
x=2, y=2
2
2
3
3
4
4
5
5
6
6
7
7 8 9
.
Page 3


A pair of linear 
equation in two 
Variables 
Substitution Method
Linear equation can be represented and solved by
Equation in the form of ax + by + c = 0, where a, b & c are real numbers and ‘a’ & ‘b’ both cannot be
zero is known as linear equation in two variables          
Graphical Method Algebraic Method
Substitution Elimination Cross Multiplication
Cross multiplication method Elimination Method
.   We nd the value of one variable ( say y ) in
 terms of the other variable ( say x ) from either of 
 the equations.
.   Substitute the value of y in another equation
 to make it in one variable. Now we can get the
  value of x.
.   Substitute the value of obtained variable x in 
  any of the equations to get the value of ‘y ’. 
.   We make coecient of a variable in one of the 
 equations equal to the coecient of the same 
 variable in another equation by multiplying it with 
 a suitable number.
.   Add or subtract one equation from other to
 eliminate one variable.
.   Now we get an equation in one variable and  
 can solve it to get its value.
 .   Substitute the value of the obtained variable in
 any of the equations to get the value of another
 variable. 
1
1 1
2
3
2
3
4
.   Let the general from of pair of linear 
equations in two variables x & y be 
a
1
x + b
1
y + c
1
 = 0
a
2
x + b
2
y + c
2
 = 0
2 .   x & y can be found by using following formulas
b
1
c
2
 - b
2
c
1
a
1
b
2
 - a
2
b
1
c
1
a
2
 - c
2
a
1
a
1
b
2
 - a
2
b
1
y =
x =
Methods to solve pair of linear equations
A PAIR OF LINEAR EQUATIONS IN TWO VARIABLES 7
A Pair of Linear Equations in Two Variables
{ Sotution }
Y
X
1
1
(0,0)
2 x + y = 6
x + y = 4 (2, 2)
x=2, y=2
2
2
3
3
4
4
5
5
6
6
7
7 8 9
.
A PAIR OF LINEAR EQUATIONS IN TWO VARIABLES 8
If
Condition of Consistency
Number of Solutions of a Pair of Linear Equations
Pair of Linear Equations
{ Lines are intersecting }
{ Unique solution }
{ Consistent }
Y
X
1
1
(0,0)
2 m + a = 6
m + a = 4
(2, 2)
2
2
3
3
4
4
5
5
6
6
7
7 8 9
.
{Lines are parallel}
{No solution}
{Inconsistent}
Y
X 1
1
(0,0)
m + a = 6
m + a = 4
2
2
3
3
4
4
5
5
6
6
7
7 8 9
= =
a
1
a
2
b
1
b
2
c
1
c
2
{Lines are coincident}
{Many solutions}
{Consistent}
Y
X
1
1
(0,0)
2
2
3
3
4
4
5
5
6
6
7
7 8 9
2m + 2a = 8
m + a = 4
= =
a
1
a
2
b
1
b
2
c
1
c
2
a
1
a
2
=
b
1
b
2
a
1
a
2
=
b
1
b
2
Number of Solutions Relation Between Lines
(Inconsistent) No Solution
(Inconsistent) No Solution
Coincident Lines
Parallel Lines
Parallel Lines
Intersecting Lines
Intersecting Lines
(Consistent) Unique Solution
(Consistent) Many Solution
(Consistent) Unique Solution
{ Then check for }
2x + 4y = 6 4x - 8y = 12 ;
x - 3y = 1 2x - 6y = 4 ;
2p - 3q = 2 3p - 2q = 2 ;
m + 4n = 1
3
2
3m + 8n = 2 ;
-6x + 2y = 3
-3x + y = 3
;
Page 4


A pair of linear 
equation in two 
Variables 
Substitution Method
Linear equation can be represented and solved by
Equation in the form of ax + by + c = 0, where a, b & c are real numbers and ‘a’ & ‘b’ both cannot be
zero is known as linear equation in two variables          
Graphical Method Algebraic Method
Substitution Elimination Cross Multiplication
Cross multiplication method Elimination Method
.   We nd the value of one variable ( say y ) in
 terms of the other variable ( say x ) from either of 
 the equations.
.   Substitute the value of y in another equation
 to make it in one variable. Now we can get the
  value of x.
.   Substitute the value of obtained variable x in 
  any of the equations to get the value of ‘y ’. 
.   We make coecient of a variable in one of the 
 equations equal to the coecient of the same 
 variable in another equation by multiplying it with 
 a suitable number.
.   Add or subtract one equation from other to
 eliminate one variable.
.   Now we get an equation in one variable and  
 can solve it to get its value.
 .   Substitute the value of the obtained variable in
 any of the equations to get the value of another
 variable. 
1
1 1
2
3
2
3
4
.   Let the general from of pair of linear 
equations in two variables x & y be 
a
1
x + b
1
y + c
1
 = 0
a
2
x + b
2
y + c
2
 = 0
2 .   x & y can be found by using following formulas
b
1
c
2
 - b
2
c
1
a
1
b
2
 - a
2
b
1
c
1
a
2
 - c
2
a
1
a
1
b
2
 - a
2
b
1
y =
x =
Methods to solve pair of linear equations
A PAIR OF LINEAR EQUATIONS IN TWO VARIABLES 7
A Pair of Linear Equations in Two Variables
{ Sotution }
Y
X
1
1
(0,0)
2 x + y = 6
x + y = 4 (2, 2)
x=2, y=2
2
2
3
3
4
4
5
5
6
6
7
7 8 9
.
A PAIR OF LINEAR EQUATIONS IN TWO VARIABLES 8
If
Condition of Consistency
Number of Solutions of a Pair of Linear Equations
Pair of Linear Equations
{ Lines are intersecting }
{ Unique solution }
{ Consistent }
Y
X
1
1
(0,0)
2 m + a = 6
m + a = 4
(2, 2)
2
2
3
3
4
4
5
5
6
6
7
7 8 9
.
{Lines are parallel}
{No solution}
{Inconsistent}
Y
X 1
1
(0,0)
m + a = 6
m + a = 4
2
2
3
3
4
4
5
5
6
6
7
7 8 9
= =
a
1
a
2
b
1
b
2
c
1
c
2
{Lines are coincident}
{Many solutions}
{Consistent}
Y
X
1
1
(0,0)
2
2
3
3
4
4
5
5
6
6
7
7 8 9
2m + 2a = 8
m + a = 4
= =
a
1
a
2
b
1
b
2
c
1
c
2
a
1
a
2
=
b
1
b
2
a
1
a
2
=
b
1
b
2
Number of Solutions Relation Between Lines
(Inconsistent) No Solution
(Inconsistent) No Solution
Coincident Lines
Parallel Lines
Parallel Lines
Intersecting Lines
Intersecting Lines
(Consistent) Unique Solution
(Consistent) Many Solution
(Consistent) Unique Solution
{ Then check for }
2x + 4y = 6 4x - 8y = 12 ;
x - 3y = 1 2x - 6y = 4 ;
2p - 3q = 2 3p - 2q = 2 ;
m + 4n = 1
3
2
3m + 8n = 2 ;
-6x + 2y = 3
-3x + y = 3
;
Variable Reducible method
+ =
4
(x - y)
1
=  p =  q
(x - y)
1
(x + y)
3
(x + y)
2
Take
,
?
4p + 3q = 2
+ =
6
y
1
=  p
=  q
x
1
y
3
x
4
Take
Divide by x?
,
?
?
6x + 3y = 4 x?
6p + 3q = 4
?
?
+
=
2
m
7
Take
n
3
2x + 7y = 3
? = ,
1
m
1
n
x = y
Introduction to Pair of
Linear Equations
Problem Solving 
of Interesting Lines
Conditions of Consistency
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PLEASE KEEP IN MIND
?
?
?
?
 Linear means straight and a linear equation refers to an equa-
tion which when plotted on the graph gives a straight line. And it is 
true for any linear equation whether it is one variable, two variable or 
multi-variable.
 To solve linear equations, number of variables should be equal 
to the number of equations.
 If it is asked to solve linear equations and the method is not 
mentioned then, one can use any of the methods which is most suita-
ble for that particular question.
 There is a formula that shows the relation between time, speed 
and distance. Use this formula for word problems.   
Speed =
Distance
Time
There can be a variety of situations which can be mathematically represented by two equations which are not linear.
But we can eventually alter them and reduce them to a pair of linear equations.   
A PAIR OF LINEAR EQUATIONS IN TWO VARIABLES 9