INTRODUCTION
 Two linear equations in the same two variables are called
linear equation in two variables the most general form of
a pair of linear equations is
a1x+b2y +c1=0
a2x+b2y+c2=0
Where a1,a2,b1,b2,c1,c2 are real numbers ,such that a12+b12=
2
2
0 a2 +b2 =0
 A pair of linear equations in two variables can be
represented and solved by the :
1. Graphical method
2. Algebraic method
Graphical method :
the graph of the pair of linear equation in
two variables is represented by two lines :
1. If the lines intersect at a point then that point gives the
unique solutions of the two equations . In this the pair
of equations is consistent.
2. If the lines coincide then there are infinitely many
solutions – each point on the line being a solutions. In
this case the pair of linear equation is dependent (
consistent ).
3. If the lines are parallel ,then the pair of linear equation is
inconsistent.
Algebraic method :
There are several method for solving the equations by
algebraic methods
1. Substitution Method
2. Elimination Method
3. Cross Multiplication Method
Two or more different linear
equations in two variables having the
same variables taken together from a
system of linear equations in two
variables. Thus a system of linear
equation is a collection or set of all
those linear equations which we deal
together simultaneously.
When the simultaneous solution are satisfied with
same pair of values of two variables, then such pair
of values of the two variables then such pair of
values of the variables is called the Solution of
simultaneous equations
Let 2x + y = 8 and x+ 3y = 9be the given
simultaneous equation. The solution of these
simultaneous equations is the pair of the x and y
that satisfy both the equation simultaneously
1.  Graphical
Method
The graphical method is the method in which the linear
equations are solved on the graph
Consider these are simultaneous equations
1) x + 2y = 0 and 3x + 4y – 20 = 0
2) 2x + 3y - 9 = 0 and 4x + 6y - 18 = 0
3) x + 2y - 4 =0 and 2x + 4y -12 =0
There are three conditions after solving the linear
equation through graphical method we get
1) Intersecting Lines
2) Coincident Lines
3) Parallel Lines
Example in the above three equations
1) The lines will Intersect
2) The lines will coincide
3) The lines will parallel
The graphs of equation x + y = 3 and x –
y = 1 are two distinct the point lines
which intersect each other in exactly one
point ( 2 , 1 ). This point of intersection
has value of x = 1 and y = 1. These
equation satisfy both the equations and
hence ( 2 , 1 ) is the solution of the given
simultaneous equations. Therefore , the coordinates of the points of intersection of
lines represented by these equations in
the solution.
4
3
2
( 2 , 1)
1
0
-5
-4
-3
-2
-1
0
-1
-2
-3
-4
-5
1
2
3
4
The graphs of the equation x - y = -2 and
2x - 2y = -4 appears to in one lines but
these are two lines drawn one above the
other here. Here every point of the line is
the intersection point. Hence all the
point on the line has solutions. Thus
these have infinitely many solutions
these lines are overlapping ( coincident ).
If the lines coincide with each other then
they have infinite solutions , each point
on the lines being a solution.
5
4
3
2
1
0
-5
-4
-3
-2
-1
0
-1
-2
-3
-4
-5
1
2
3
4
5
Consider the equation 2x - y = -1
and 2x – y = -4. Do they have a
common solution the lines. These
equations have no solution. The
line of these two equations does
not intersect each other.
Therefore the lines represented
by these equations are parallel.
5
4
3
2
1
0
-5
-4
-3
-2
-1
0
-1
-2
-3
-4
-5
1
2
3
4
5
Limitations of graphical
method
2.1 Elimination Method
Elimination method is the second method of solving linear
equation by algebraic method. This method is sometime
more convenient than the substitution method
Step 1 :- First multiply both the equation by some suitable
non-zero constants to make the co efficient of one variable
either x or y
Step 2 :- Then add or subtract both equation so that the on
variable get eliminated if you get one variable go to step 3
If in step 2 if we obtain a true statement involving no
variable then the original pair of linear equation have
infinitely many solution
If in step 2 we obtain a false statement involving no
variable then the original pair linear equation have no
solution
Step 3 :- solve the equation in one variable x or y so
obtained to get its value
Step 4 :- Substitute the value of x in the either of original
equations to get the value of the other variable
Example :-
x +y=5
1)
2x – 3y = 4
2)
We will multiply equation 1) by 2
After multiplication of equation 1 we got the equation
2x + 2y = 10
3)
Subtraction of equation 2) and 3)
2x + 2y = 10
2x – 3y = 4
5y = 6
y=
By putting value of y in equation 1)
x+y=5
x+
=5
x= 5x=
x=
This one of the algebraic method to solve
simultaneous equation in this method the
equations are solved by taking value of
each variable separately let s see the steps
to solve the Substitution method
These are the steps to solve linear equation
( simultaneous equation ) by substitution
method
Step 1:- find the value of one variable, say
y in terms of other , i.e. x from either
equation whichever is convenient.
Step 2:- substitute the value of y in the
other equation and reduce it to an
equation in one variable , i.e. the terms of
x which can be solved
Step 3:- Substitute the value of x or y
obtained in step 2 in the equation used in
Step 1 to obtain the value of the other
variable.
Example :- Let the two equations be
3x + 4y = 10 ; x - y = 1
x–y =1
x=y+1
x-y=1
x=y+1
3x + 4y = 10
1)
2)
3)
3 ( y + 1 ) + 4y = 10
4)
3y + 3 + 4y = 10
3y + 4y + 3 = 10
7y + 3 = 10
7y = 10 – 3
7y = 7
y=
y=1
x=y+1
x=1+1
x=2
Here is the answer x = 2 ; y = 1
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
x
b1
b2
y
c1
c2
1
a1
a2
b1
b2
For solving the method of cross –
multiplication method we have to
follow then following steps :Step 1 :- Write the equations in the
form 1) & 2)
Step 2 :- take the help of the above
diagram and equation written above
Step 3:- Find x and y provided a1b2 a2b1 ≠ 0
Gabriel Cramer
Gabriel Cramer was born on 31 July 1704
in Geneva and dead on 4 January 1752. He
was a Swiss Mathematician. He showed
excellence in mathematics from an early
age. At
the age of 18 he received his
doctorate. In 1728 he proposed the solution
to St. Petersburg Paradox. He edited the
works of two elders Bernoullis and wrote on
the physical cause of the spheriodial shape of
the planets in 1730 and on the Newton’s
treatment of cubic curves in 1956. he was
professor at Geneva. He has proposed the
algebraic method for solving linear equation
in two variables named cross-multiplication
or Cramer’s rule.
Summary
 Linear equations in two variables have infinite
solutions
 Linear equations in two variables is expressed as
ax + by + c = 0 where a ≠ 0 b ≠ 0
 Two or more linear equation having same variables
taken form linear equation in two variables .
When we consider linear equation in two variables
When we consider linear equation in two
variables such equation are called Simultaneous
Equation
When the Simultaneous equations are
satisfied by the same pair of values of the two
variables then such pair of values of variables
is called the solutions of given Simultaneous
Equation
Methods to solve Simultaneous Equation
Graphical method
Algebraic method
The Elimination Method
Substitution Method
Cross - Multiplication Method
The Simultaneous Equation are said to be
consistent if they have solutions

The Simultaneous Equation are said to be
inconsistent if they have no solutions