Order of presentation Anushka - Opening Nikunj -Intro Shubham - Graphical Amel - Sunstitution Siddhartha- Elimination Karthik - Cross multiplication Anushka - Equations reducible...& wrap-up In case of any confusion..inform me by facebook, phone or in school

1. PAIR OF LINEAR EQUATIONS
IN TWO VARIABLE

2. INTRODUCTION TO
PAIR OF LINEAR EQUATION IN TWO
VARIABLES
A pair of linear equation is said to form a
system of simultaneous linear equation in the
standard form
a1x+b1y+c1=0
a2x+b2y+c2=0
Where ‘a’, ‘b’ and ‘c’ are not equal to real
numbers ‘a’ and ‘b’ are not equal to zero.

3. DERIVING THE SOLUTION THROUGH
GRAPHICAL METHOD
Let us consider the following system of two
simultaneous linear equations in two variable.
2x – y = -1 ;3x + 2y = 9
We can determine the value of the a variable by
substituting any value for the other variable, as done
in the given examples
X 0 2
Y 1 5
X 3 -1
Y 0 6
X=(y-1)/2 y=2x+1 2y=9-3x x=(9-2y)/3
2x – y = -1 3x + 2y = 9

4. 6
5
4
3
2
1
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5
-1
-2
-3
-4
-5
(1,3)
(2,5)
(-1,6)
(0,1)
X=1 Y=3
X 0 2
Y 1 5
X 3 -1
Y 0 6
EQUATION 1
EQUATION 2
‘X’ intercept = 1 ‘Y’ intercept = 3
2x – y = -1
3x + 2y = 9

5. ax1 + by1 + c1 = 0;
ax2 + by2 + c2 = 0
a1 b1 c1
c2a2 b2
=i)
ii)
iii)
=
a1 b1
a2 b2
=
a1 b1 c1
c2a2 b2
==
Intervening Lines; Infinite
Solutions
Intersecting Lines; Definite
Solution
Parallel Lines; No Solution

6. DERIVING THE SOLUTION THROUGH
SUBSTITUTION METHOD
This method involves substituting the value of
one variable, say x , in terms of the other in
the equation to turn the expression into a
Linear Equation in one variable, in order to
derive the solution of the equation .
For example
x + 2y = -1 ;2x – 3y = 12

7. 2x – 3y = 12 ----------(ii)
x = -2y -1
x = -2 x (-2) – 1
= 4–1
x = 3
x + 2y = -1 -------- (i)
x + 2y = -1
x = -2y -1 ------- (iii)
Substituting the value of x
inequation (ii), we get
2x – 3y = 12
2 ( -2y – 1) – 3y
= 12 - 4y – 2 – 3y
= 12 - 7y = 14
= 12 - 14 = 7y
y = -2
Putting the value of y
in eq. (iii), we get
Hence the solution of the equation is ( 3, - 2 )

8. DERIVING THE SOLUTION THROUGH
ELIMINATION METHOD
In this method, we eliminate one of the two variables
to obtain an equation in one variable which can
easily be solved. The value of the other variable can
be obtained by putting the value of this variable in
any of the given equations.
For example:
3x + 2y = 11 ;2x + 3y = 4

9. 3x + 2y = 11 --------- (i) 2x + 3y = 4 ---------(ii)
3x + 2y = 11 x3-
9x - 3y = 33---------(iii)
=>9x + 6y = 33-----------(iii)
4x + 6y = 8------------(iv)
(-) (-) (-)
(iii) – (iv) =>
x3 2x + 3y = 4
4x + 6y = 8---------(ii)
x2
5x = 25
x = 5
Putting the value of x in
equation (ii) we get, =>
2x + 3y = 4
2 x 5 + 3y = 4
10 + 3y = 4
3y = 4 – 10
3y = - 6
y=-2
Hence, x = 5 and y = -2

10. DERIVING THE SOLUTION THROUGH
CROSS-MULTIPLICATION METHOD
The method of obtaining solution of simultaneous equation by
using determinants is known as Cramer’s rule. In this method we
have to follow this equation and diagram
ax1 + by1 + c1 = 0;
ax2 + by2 + c2 = 0
b1c2 –b2c1
a1b2 –a2b1
c1a2 –c2a1
a1b2 –a2b1
X= Y=

11. X
B1c2-b2c1
Y
c1a2 –c2a1
=
1
a1b2 –a2b1
=
b1c2 –b2c1
a1b2 –a2b1
c1a2 –c2a1
a1b2 –a2b1
X= Y=

12. Example:
8x + 5y – 9 = 0 3x + 2y – 4 = 0
X
-20-(-18)
Y
-27-(-32)
=
1
16-15
=
X Y 1
1-2 5
=
X
-2
Y
5
=1 1
X = -2 and Y = 5
X
B1c2-b2c1
Y
c1a2 –c2a1
=
1
a1b2 –a2b1
=

13. EQUATIONS REDUCIBLE TO PAIR OF
LINEAR EQUATION IN TWO VARIABLES
In case of equations which are not linear, like
We can turn the equations into linear equations by
substituting
2 3
13
x y
=
5 4
-2
x y
=+ -
1
p
x
=
1
q
y
=

14. The resulting equations are
2p + 3q = 13 ; 5p - 4q = -2
These equations can now be solved by any of
the aforementioned methods to derive the
value of ‘p’ and ‘q’.
‘p’ = 2 ;‘q’ = 3
We know that
1
p
x
=
1
q
y
=
1
X
2
=
1
Y
3
=
&

15. SUMMARY
• Insight to Pair of Linear Equations in Two Variable
• Deriving the value of the variable through
• Graphical Method
• Substitution Method
• Elimination Method
• Cross-Multiplication Method
• Reducing Complex Situation to a Pair of Linear
Equations to derive their solution