01 - Let's Recap - Pair of linear equations in Two Variables - Class 10 - Maths Class 10 Notes | EduRev

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Class 10 : 01 - Let's Recap - Pair of linear equations in Two Variables - Class 10 - Maths Class 10 Notes | EduRev

 Page 1


  
 
PAIR OF LINEAR EQUATIONS IN TWO VARIABLES 
 
 
Algebraic Expression: A combination of constants and variables, connected by 
four fundamental arithmetic operations of +, -, x and ÷ is called an algebraic 
expression. 
 
Equation: An algebraic expression with equal to sign (=) is called an equation. 
Without an equal to sign, it is considered as an expression only. 
 
Linear Equation: If the greatest exponent of the variable(s) in an equation is one, 
then the equation is said to be a linear equation. 
 
If number of variable used in linear equation is one, then equation is said to be 
linear equation in two variables. 
 
Linear Equation in Two Variables: 
An equation which can be put in the form 0 = + + c by ax , where a, b and c are real 
numbers, and 0 , 0 ? ? b a
 
is called a linear equation in two variables x and y
.
 
 
General Form of a Pair of Linear Equation in Two Variables: 
 
 A pair of linear equations in two variables x and y can be represented as 
follows: 
     a
1
x + b
1
y + c
1
 = 0 
  
     a
2
x + b
2
y + c
2
 = 0,  
 
     where a
1
, a
2
, b
1
, b
2
, c
1
, c
2
 are real numbers such that a
1
2
 + b
1
2
 ? 0 , a
2
2
 + b
2
2
 ? 0 
 
 System of simultaneous linear equations: 
 
Consistent system: A system of simultaneous linear equations is said to be 
consistent if it has at least one solution. 
 
Inconsistent system: A system of simultaneous linear equations is said to be 
inconsistent if it has no solution. 
 
Page 2


  
 
PAIR OF LINEAR EQUATIONS IN TWO VARIABLES 
 
 
Algebraic Expression: A combination of constants and variables, connected by 
four fundamental arithmetic operations of +, -, x and ÷ is called an algebraic 
expression. 
 
Equation: An algebraic expression with equal to sign (=) is called an equation. 
Without an equal to sign, it is considered as an expression only. 
 
Linear Equation: If the greatest exponent of the variable(s) in an equation is one, 
then the equation is said to be a linear equation. 
 
If number of variable used in linear equation is one, then equation is said to be 
linear equation in two variables. 
 
Linear Equation in Two Variables: 
An equation which can be put in the form 0 = + + c by ax , where a, b and c are real 
numbers, and 0 , 0 ? ? b a
 
is called a linear equation in two variables x and y
.
 
 
General Form of a Pair of Linear Equation in Two Variables: 
 
 A pair of linear equations in two variables x and y can be represented as 
follows: 
     a
1
x + b
1
y + c
1
 = 0 
  
     a
2
x + b
2
y + c
2
 = 0,  
 
     where a
1
, a
2
, b
1
, b
2
, c
1
, c
2
 are real numbers such that a
1
2
 + b
1
2
 ? 0 , a
2
2
 + b
2
2
 ? 0 
 
 System of simultaneous linear equations: 
 
Consistent system: A system of simultaneous linear equations is said to be 
consistent if it has at least one solution. 
 
Inconsistent system: A system of simultaneous linear equations is said to be 
inconsistent if it has no solution. 
 
  
 PAIR OF LINEAR EQUATIONS IN TWO VARIABLES 
 
 
 A pair of linear equations in two variables can be solved by the: 
 
           (i)  Graphical Method. 
           (ii) Algebraic Method. 
     Algebraic methods are of three types: 
(a) Substitution Method. 
            (b) Elimination Method.  
            (c) Cross-multiplication Method. 
 
Graphical Method: 
  
      (i) Intersecting, if    


?


 
 
       here, the equations have a unique solution, and pair of equations is said to be  
       consistent. 
      (ii) Parallel, if 


=


?


 
 
       here, the equations have no solution, and pair of equations is said to be   
       inconsistent. 
       (iii) Coincident, if 


=


=


       
 
        here, the equations have infinitely many solutions, and pair of equations is  
        said to be consistent. 
 
Algebraic Method: 
 
       a) Substitution Method: 
          Let us consider any two equations 
        
) ( .......... 2
2
A
b
y
a
x
= +
 
 
        
) ( .......... 4 B
b
y
a
x
= - 
Page 3


  
 
PAIR OF LINEAR EQUATIONS IN TWO VARIABLES 
 
 
Algebraic Expression: A combination of constants and variables, connected by 
four fundamental arithmetic operations of +, -, x and ÷ is called an algebraic 
expression. 
 
Equation: An algebraic expression with equal to sign (=) is called an equation. 
Without an equal to sign, it is considered as an expression only. 
 
Linear Equation: If the greatest exponent of the variable(s) in an equation is one, 
then the equation is said to be a linear equation. 
 
If number of variable used in linear equation is one, then equation is said to be 
linear equation in two variables. 
 
Linear Equation in Two Variables: 
An equation which can be put in the form 0 = + + c by ax , where a, b and c are real 
numbers, and 0 , 0 ? ? b a
 
is called a linear equation in two variables x and y
.
 
 
General Form of a Pair of Linear Equation in Two Variables: 
 
 A pair of linear equations in two variables x and y can be represented as 
follows: 
     a
1
x + b
1
y + c
1
 = 0 
  
     a
2
x + b
2
y + c
2
 = 0,  
 
     where a
1
, a
2
, b
1
, b
2
, c
1
, c
2
 are real numbers such that a
1
2
 + b
1
2
 ? 0 , a
2
2
 + b
2
2
 ? 0 
 
 System of simultaneous linear equations: 
 
Consistent system: A system of simultaneous linear equations is said to be 
consistent if it has at least one solution. 
 
Inconsistent system: A system of simultaneous linear equations is said to be 
inconsistent if it has no solution. 
 
  
 PAIR OF LINEAR EQUATIONS IN TWO VARIABLES 
 
 
 A pair of linear equations in two variables can be solved by the: 
 
           (i)  Graphical Method. 
           (ii) Algebraic Method. 
     Algebraic methods are of three types: 
(a) Substitution Method. 
            (b) Elimination Method.  
            (c) Cross-multiplication Method. 
 
Graphical Method: 
  
      (i) Intersecting, if    


?


 
 
       here, the equations have a unique solution, and pair of equations is said to be  
       consistent. 
      (ii) Parallel, if 


=


?


 
 
       here, the equations have no solution, and pair of equations is said to be   
       inconsistent. 
       (iii) Coincident, if 


=


=


       
 
        here, the equations have infinitely many solutions, and pair of equations is  
        said to be consistent. 
 
Algebraic Method: 
 
       a) Substitution Method: 
          Let us consider any two equations 
        
) ( .......... 2
2
A
b
y
a
x
= +
 
 
        
) ( .......... 4 B
b
y
a
x
= - 
  
       PAIR OF LINEAR EQUATIONS IN TWO VARIABLES 
 
 
 Get the value of x in terms of y from equation (A), Substitute the value of x 
 in equation (B) to get value of y.   
 
b) Elimination Method: 
           
    Consider two equations 
       
) ( .......... 2
2
A
b
y
a
x
= + 
       
) ( .......... 4 B
b
y
a
x
= - 
       
     Step1: Comparing the coefficients of   x and y. 
 
                 
) .......( 2 2 i ab ay bx = +
 
 
                   
) .......( 4 ii ab ay bx = -
 
 
     Step2: Simplify the equation either by adding or subtracting. 
 
     Step 3: After simplifying you will find one value either ‘x’ or ‘y’. 
 
     Step4: Substitute value of ‘x’ or ‘y’ in any of one equation to get the value 
                 of other variable.
  
 
    
 
    c) Cross Multiplication Method: 
        Consider two equations: 
       
) ( .......... 2
2
A
b
y
a
x
= +
 
 
        
) ( .......... 4 B
b
y
a
x
= - 
    Step1: 
  
 Below ‘x’ write the coefficients of ‘y’ and the constant terms. 
 Below ‘y’ write the coefficients of ‘x’ and the constant terms. 
 Below 1, write the coefficients of ‘x’ and ‘y’. 
     
 
Page 4


  
 
PAIR OF LINEAR EQUATIONS IN TWO VARIABLES 
 
 
Algebraic Expression: A combination of constants and variables, connected by 
four fundamental arithmetic operations of +, -, x and ÷ is called an algebraic 
expression. 
 
Equation: An algebraic expression with equal to sign (=) is called an equation. 
Without an equal to sign, it is considered as an expression only. 
 
Linear Equation: If the greatest exponent of the variable(s) in an equation is one, 
then the equation is said to be a linear equation. 
 
If number of variable used in linear equation is one, then equation is said to be 
linear equation in two variables. 
 
Linear Equation in Two Variables: 
An equation which can be put in the form 0 = + + c by ax , where a, b and c are real 
numbers, and 0 , 0 ? ? b a
 
is called a linear equation in two variables x and y
.
 
 
General Form of a Pair of Linear Equation in Two Variables: 
 
 A pair of linear equations in two variables x and y can be represented as 
follows: 
     a
1
x + b
1
y + c
1
 = 0 
  
     a
2
x + b
2
y + c
2
 = 0,  
 
     where a
1
, a
2
, b
1
, b
2
, c
1
, c
2
 are real numbers such that a
1
2
 + b
1
2
 ? 0 , a
2
2
 + b
2
2
 ? 0 
 
 System of simultaneous linear equations: 
 
Consistent system: A system of simultaneous linear equations is said to be 
consistent if it has at least one solution. 
 
Inconsistent system: A system of simultaneous linear equations is said to be 
inconsistent if it has no solution. 
 
  
 PAIR OF LINEAR EQUATIONS IN TWO VARIABLES 
 
 
 A pair of linear equations in two variables can be solved by the: 
 
           (i)  Graphical Method. 
           (ii) Algebraic Method. 
     Algebraic methods are of three types: 
(a) Substitution Method. 
            (b) Elimination Method.  
            (c) Cross-multiplication Method. 
 
Graphical Method: 
  
      (i) Intersecting, if    


?


 
 
       here, the equations have a unique solution, and pair of equations is said to be  
       consistent. 
      (ii) Parallel, if 


=


?


 
 
       here, the equations have no solution, and pair of equations is said to be   
       inconsistent. 
       (iii) Coincident, if 


=


=


       
 
        here, the equations have infinitely many solutions, and pair of equations is  
        said to be consistent. 
 
Algebraic Method: 
 
       a) Substitution Method: 
          Let us consider any two equations 
        
) ( .......... 2
2
A
b
y
a
x
= +
 
 
        
) ( .......... 4 B
b
y
a
x
= - 
  
       PAIR OF LINEAR EQUATIONS IN TWO VARIABLES 
 
 
 Get the value of x in terms of y from equation (A), Substitute the value of x 
 in equation (B) to get value of y.   
 
b) Elimination Method: 
           
    Consider two equations 
       
) ( .......... 2
2
A
b
y
a
x
= + 
       
) ( .......... 4 B
b
y
a
x
= - 
       
     Step1: Comparing the coefficients of   x and y. 
 
                 
) .......( 2 2 i ab ay bx = +
 
 
                   
) .......( 4 ii ab ay bx = -
 
 
     Step2: Simplify the equation either by adding or subtracting. 
 
     Step 3: After simplifying you will find one value either ‘x’ or ‘y’. 
 
     Step4: Substitute value of ‘x’ or ‘y’ in any of one equation to get the value 
                 of other variable.
  
 
    
 
    c) Cross Multiplication Method: 
        Consider two equations: 
       
) ( .......... 2
2
A
b
y
a
x
= +
 
 
        
) ( .......... 4 B
b
y
a
x
= - 
    Step1: 
  
 Below ‘x’ write the coefficients of ‘y’ and the constant terms. 
 Below ‘y’ write the coefficients of ‘x’ and the constant terms. 
 Below 1, write the coefficients of ‘x’ and ‘y’. 
     
 
 
PAIR OF LINEAR EQUATIONS IN TWO VARIABLES
 
   Step2: 
 Simplify the equations.
    Step3: 
 After simplifying
we get value of ‘x’ and ‘y’.
 
 
Equation Reducible to a
 
There are several situations which can be mathematically 
equations that are not linear to start with. But we a
of linear equations. 
 
We will be easily able to understand this concept of equation reducible to a pair of 
linear equation in two variables..
 
We shall discuss the solution of such pairs of equations which are not linear but 
can be reduced to linear form by making some suitable substitutions. We now 
explain this process through some examples.
Example: Solve the pair of equations:
 
 
 
Solution: Let us write the given pair of equations as
 
 
 
 
 
 
PAIR OF LINEAR EQUATIONS IN TWO VARIABLES
the equations. 
After simplifying equate the values of ‘x’   and ‘y’ with constant terms, 
get value of ‘x’ and ‘y’. 
a Pair of Linear Equations In Two Variables
There are several situations which can be mathematically represented by two 
equations that are not linear to start with. But we alter them so as to 
We will be easily able to understand this concept of equation reducible to a pair of 
linear equation in two variables.. 
shall discuss the solution of such pairs of equations which are not linear but 
can be reduced to linear form by making some suitable substitutions. We now 
explain this process through some examples. 
the pair of equations: 
us write the given pair of equations as 
PAIR OF LINEAR EQUATIONS IN TWO VARIABLES 
and ‘y’ with constant terms, 
f Linear Equations In Two Variables 
represented by two 
lter them so as to reduce to pair 
We will be easily able to understand this concept of equation reducible to a pair of 
shall discuss the solution of such pairs of equations which are not linear but 
can be reduced to linear form by making some suitable substitutions. We now 
Page 5


  
 
PAIR OF LINEAR EQUATIONS IN TWO VARIABLES 
 
 
Algebraic Expression: A combination of constants and variables, connected by 
four fundamental arithmetic operations of +, -, x and ÷ is called an algebraic 
expression. 
 
Equation: An algebraic expression with equal to sign (=) is called an equation. 
Without an equal to sign, it is considered as an expression only. 
 
Linear Equation: If the greatest exponent of the variable(s) in an equation is one, 
then the equation is said to be a linear equation. 
 
If number of variable used in linear equation is one, then equation is said to be 
linear equation in two variables. 
 
Linear Equation in Two Variables: 
An equation which can be put in the form 0 = + + c by ax , where a, b and c are real 
numbers, and 0 , 0 ? ? b a
 
is called a linear equation in two variables x and y
.
 
 
General Form of a Pair of Linear Equation in Two Variables: 
 
 A pair of linear equations in two variables x and y can be represented as 
follows: 
     a
1
x + b
1
y + c
1
 = 0 
  
     a
2
x + b
2
y + c
2
 = 0,  
 
     where a
1
, a
2
, b
1
, b
2
, c
1
, c
2
 are real numbers such that a
1
2
 + b
1
2
 ? 0 , a
2
2
 + b
2
2
 ? 0 
 
 System of simultaneous linear equations: 
 
Consistent system: A system of simultaneous linear equations is said to be 
consistent if it has at least one solution. 
 
Inconsistent system: A system of simultaneous linear equations is said to be 
inconsistent if it has no solution. 
 
  
 PAIR OF LINEAR EQUATIONS IN TWO VARIABLES 
 
 
 A pair of linear equations in two variables can be solved by the: 
 
           (i)  Graphical Method. 
           (ii) Algebraic Method. 
     Algebraic methods are of three types: 
(a) Substitution Method. 
            (b) Elimination Method.  
            (c) Cross-multiplication Method. 
 
Graphical Method: 
  
      (i) Intersecting, if    


?


 
 
       here, the equations have a unique solution, and pair of equations is said to be  
       consistent. 
      (ii) Parallel, if 


=


?


 
 
       here, the equations have no solution, and pair of equations is said to be   
       inconsistent. 
       (iii) Coincident, if 


=


=


       
 
        here, the equations have infinitely many solutions, and pair of equations is  
        said to be consistent. 
 
Algebraic Method: 
 
       a) Substitution Method: 
          Let us consider any two equations 
        
) ( .......... 2
2
A
b
y
a
x
= +
 
 
        
) ( .......... 4 B
b
y
a
x
= - 
  
       PAIR OF LINEAR EQUATIONS IN TWO VARIABLES 
 
 
 Get the value of x in terms of y from equation (A), Substitute the value of x 
 in equation (B) to get value of y.   
 
b) Elimination Method: 
           
    Consider two equations 
       
) ( .......... 2
2
A
b
y
a
x
= + 
       
) ( .......... 4 B
b
y
a
x
= - 
       
     Step1: Comparing the coefficients of   x and y. 
 
                 
) .......( 2 2 i ab ay bx = +
 
 
                   
) .......( 4 ii ab ay bx = -
 
 
     Step2: Simplify the equation either by adding or subtracting. 
 
     Step 3: After simplifying you will find one value either ‘x’ or ‘y’. 
 
     Step4: Substitute value of ‘x’ or ‘y’ in any of one equation to get the value 
                 of other variable.
  
 
    
 
    c) Cross Multiplication Method: 
        Consider two equations: 
       
) ( .......... 2
2
A
b
y
a
x
= +
 
 
        
) ( .......... 4 B
b
y
a
x
= - 
    Step1: 
  
 Below ‘x’ write the coefficients of ‘y’ and the constant terms. 
 Below ‘y’ write the coefficients of ‘x’ and the constant terms. 
 Below 1, write the coefficients of ‘x’ and ‘y’. 
     
 
 
PAIR OF LINEAR EQUATIONS IN TWO VARIABLES
 
   Step2: 
 Simplify the equations.
    Step3: 
 After simplifying
we get value of ‘x’ and ‘y’.
 
 
Equation Reducible to a
 
There are several situations which can be mathematically 
equations that are not linear to start with. But we a
of linear equations. 
 
We will be easily able to understand this concept of equation reducible to a pair of 
linear equation in two variables..
 
We shall discuss the solution of such pairs of equations which are not linear but 
can be reduced to linear form by making some suitable substitutions. We now 
explain this process through some examples.
Example: Solve the pair of equations:
 
 
 
Solution: Let us write the given pair of equations as
 
 
 
 
 
 
PAIR OF LINEAR EQUATIONS IN TWO VARIABLES
the equations. 
After simplifying equate the values of ‘x’   and ‘y’ with constant terms, 
get value of ‘x’ and ‘y’. 
a Pair of Linear Equations In Two Variables
There are several situations which can be mathematically represented by two 
equations that are not linear to start with. But we alter them so as to 
We will be easily able to understand this concept of equation reducible to a pair of 
linear equation in two variables.. 
shall discuss the solution of such pairs of equations which are not linear but 
can be reduced to linear form by making some suitable substitutions. We now 
explain this process through some examples. 
the pair of equations: 
us write the given pair of equations as 
PAIR OF LINEAR EQUATIONS IN TWO VARIABLES 
and ‘y’ with constant terms, 
f Linear Equations In Two Variables 
represented by two 
lter them so as to reduce to pair 
We will be easily able to understand this concept of equation reducible to a pair of 
shall discuss the solution of such pairs of equations which are not linear but 
can be reduced to linear form by making some suitable substitutions. We now 
  
PAIR OF LINEAR EQUATIONS IN TWO VARIABLES 
 
These equations are not in the form ax + by + c = 0. However, if we  
Substitute  


  = p  and  

	
  = q in Equations (1) and (2)  we get, 
2p + 3q = 13   ---------------------------------------   (3) 
 
5p - 4q  = -2   ----------------------------------------  (4) 
So, we have expressed the equations as a pair of linear equations. Now, you can 
use any method to solve these equations, and get p = 2 and q = 3. 
We know that  p =  


 and q = 

	
 
Substitute the values of p and q to get  


 = 2 i.e. x = 


 and 

	
 = 3 i.e. y = 



 
Verification: By substituting x = 


   and y = 



  in the given equations, we find that 
both the equations are satisfied. 
 
 
 
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