10th CBSE MATHS FORMULAE CHAPTER NO.3: PAIR OF LINEAR EQUATIONS IN TWO VARIABLES

CHAPTER NO.3 Pair of Linear Equation in two variable

1)General form of linear equation: 

1) In one variable;

ax+b=0 (a≠ 0)

2) In two variable:

ax+by+c=0

2) General form of pairs of linear equation in two variables:

a₁x+b₁y+c₁=0   and  a₂x+b₂y+c₂=0

• Geometrical interpretation and algebraic interpretation
  

Pair of Linear Equation	Algebraic Condition	Graphical  Interpretation	Algebraic  Interpretation	Consistency
a₁x+b₁y+c₁=0                       and    a₂x+b₂y+c₂=0	a₁/a₂ ≠ b₁/b₂	Intersecting  lines	Unique Solution	Consistency
	a₁/a₂=b₁/b₂=c₁/c₂	Coincident lines	Infinitely many solution	Dependent    consistency
	a₁/a₂=b₁/b₂≠c₁/c₂	Parallel lines	No solution	Inconsistent

#Methods of Solving Pair of Linear Equations:

(A) Graphical method:

Step 1: Plot the both same linear equations on the same graph.

Step 2: 1) If Lines are intersecting then there are unique solution then  a₁/a₂ ≠ b₁/b₂ plot graph as shown in below graph (Fig. 3.1)

2)If Lines are Coincident then  there are  Infinitely many solution a₁/a₂=b₁/b₂=c₁/c₂ plot the graph as shown in below graph (Fig.3.2)

3)If lines are Parallel lines then there are No solution then a₁/a₂=b₁/b₂≠c₁/c₂ plot graph as shown in below graph (Fig.3.3)

(B) Algebraic Method:

      (ⅰ) Substitution method

      (ⅱ) Elimination method

      (ⅲ) Cross-Multiplication method

(i) substitution method

 a₁x+b₁y+c₁=0  and a₂x+b₂y+c₂=0

Ex. x+y=14    𑁋①

     x-y=4      𑁋②

Step 1:- write one variable x in terms of other variable y 

let consider equation ①

x=14-y

Step 2:- Substitute x in eq ②

     14-y-y=4

     14-2y=4

         2y=10

y=5

Step 3:- put y=5 in eq①
                x=14-5

x=9

(ii) Elimination method

 a₁x+b₁y+c₁=0  and a₂x+b₂y+c₂=0 

Example:- 2x+y=14    𑁋①

                 x-y=2        𑁋②

Step 1:- Multiply eqn ② by 2 and make coefficient of x equal  

             Now,
              2x+y=14𑁋③
              2x-2y=4𑁋④

Step 2:- Substract eqn ③ and ④ to eliminate x, because coefficient of y are same 
              we get 
                    (2x+y)-(2x-2y)=10
                          3y=10
                           y=10/3

Step 3:- Put y=10/3 in eqn ②
             x-10/3=2
            x=16/3

(iii) Cross multiplication method 

   For a pair of linear equations, 
      a₁x+b₁y+c₁=0  and a₂x+b₂y+c₂=0