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Presented By:- Abhinav Somani
System Of Equations
• A pair of Linear equation
in two variables is said to
form a system of linear
equation.
• For Example:-
 2x-3y+9=0
 x+4y+2=0
• These Equations form a
system of Two Linear
Equations in variables x
and y.
General form of linear equations in
two variables
• The general form of a linear
equation in two variables x and y
is ax + by + c = 0, a and b is not
equal to zero, where a, b and c
being real numbers.
• A solution of such an equation is
a pair of values, One for x and
the other for y, Which makes
two sides of the equation equal.
Every linear equation in two
variables has infinitely many
solutions which can be
represented on a certain line.
Graphical Solutions of a Linear
Equations
• Let us consider the
following system of two
simultaneous linear
equations in two variable.
 2x+y-8=0
 2x-5y+4=0
• Here we assign any value
to one of the two
variables and then
determine the value of
the other variable from
the given equation
Solving Equations
1. 2x+y-8=0
X 4 0
Y 0 8
2. 2x-5y+4=0
X -2 0
Y 0 4/5
• 2x+y=8
2x+0=8
x=8/4
x=2
• 2x+y=8
2x0+y=8
0+y=8
y=8
• 2x-5y= -4
2x-5x0= -4
2x-0= -4
x= -2
• 2x-5y= -4
2x0-5y= -4
0-5y= -4
y=4/5
Algebraic Methods of Solving
Simultaneous Linear Equations
• The most commonly used algebraic methods
of solving simultaneous linear equations in
two variables are:-
 Method of Elimination by substitution
 Method of Elimination by equating the
coefficient.
Method of Cross Multiplication.
Elimination by substitution
• Obtain the two equations. Let the equations be
a1x+b1y+c1=0 (i)
a2x+b2y+c2=0 (ii)
• Choose either of the two equations say (i) and
find the value of one variable say ‘y’ in terms of
x.
• Substitute the value of y, obtained in the
previous step in equation (ii) to get an equation
in x.
• Solve the equation obtained in the previous
step to get the value of x.
• Substitute the value of x and get the value of y.
Let us take an example
x+2y=-1 (i)
2x-3y=12 (ii)
 For (i) equation
1. x+2y=-1
x=-2y-1 (iii)
• Substituting the value of x in equation (ii),
we get
2x-3y=12
2(-2y-1) -3y=12
-4y-2-3y=12
-4y-3y=12+2
-7y=14
y=-2
• Putting the value of y in equation (iii) we get
x=-2(-2)-1
x= 4-1
x=3
Elimination Method
• In this method we eliminate one of the two
variables to obtain an equation in one variable
which can easily be solved.
• Putting the value of the variable in any of the
given equations. The value of the other
variable can be obtained.
• For example:-
3x+2y=11
2x+3y=4
• Let 3x+2y=11 (i)
2x+3y=4 (ii)
• Multiply 3 in equation (i) and 2 in equation (ii)
and subtracting equation (iv) from (iii), we get
9x+6y=33 (iii)
4x+6y=8 (iv)
5x =25
x=5
• Putting the value of x in equation (ii) we get,
2x+3y=4
2x5+3y=4
10+3y=4
3y=4-10
3y=-6
y=-2

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Linear equations in Two Variable

  • 2. System Of Equations • A pair of Linear equation in two variables is said to form a system of linear equation. • For Example:-  2x-3y+9=0  x+4y+2=0 • These Equations form a system of Two Linear Equations in variables x and y.
  • 3. General form of linear equations in two variables • The general form of a linear equation in two variables x and y is ax + by + c = 0, a and b is not equal to zero, where a, b and c being real numbers. • A solution of such an equation is a pair of values, One for x and the other for y, Which makes two sides of the equation equal. Every linear equation in two variables has infinitely many solutions which can be represented on a certain line.
  • 4. Graphical Solutions of a Linear Equations • Let us consider the following system of two simultaneous linear equations in two variable.  2x+y-8=0  2x-5y+4=0 • Here we assign any value to one of the two variables and then determine the value of the other variable from the given equation
  • 5. Solving Equations 1. 2x+y-8=0 X 4 0 Y 0 8 2. 2x-5y+4=0 X -2 0 Y 0 4/5 • 2x+y=8 2x+0=8 x=8/4 x=2 • 2x+y=8 2x0+y=8 0+y=8 y=8 • 2x-5y= -4 2x-5x0= -4 2x-0= -4 x= -2 • 2x-5y= -4 2x0-5y= -4 0-5y= -4 y=4/5
  • 6.
  • 7. Algebraic Methods of Solving Simultaneous Linear Equations • The most commonly used algebraic methods of solving simultaneous linear equations in two variables are:-  Method of Elimination by substitution  Method of Elimination by equating the coefficient. Method of Cross Multiplication.
  • 8. Elimination by substitution • Obtain the two equations. Let the equations be a1x+b1y+c1=0 (i) a2x+b2y+c2=0 (ii) • Choose either of the two equations say (i) and find the value of one variable say ‘y’ in terms of x. • Substitute the value of y, obtained in the previous step in equation (ii) to get an equation in x.
  • 9. • Solve the equation obtained in the previous step to get the value of x. • Substitute the value of x and get the value of y. Let us take an example x+2y=-1 (i) 2x-3y=12 (ii)  For (i) equation 1. x+2y=-1 x=-2y-1 (iii)
  • 10. • Substituting the value of x in equation (ii), we get 2x-3y=12 2(-2y-1) -3y=12 -4y-2-3y=12 -4y-3y=12+2 -7y=14 y=-2
  • 11. • Putting the value of y in equation (iii) we get x=-2(-2)-1 x= 4-1 x=3
  • 12. Elimination Method • In this method we eliminate one of the two variables to obtain an equation in one variable which can easily be solved. • Putting the value of the variable in any of the given equations. The value of the other variable can be obtained. • For example:- 3x+2y=11 2x+3y=4
  • 13. • Let 3x+2y=11 (i) 2x+3y=4 (ii) • Multiply 3 in equation (i) and 2 in equation (ii) and subtracting equation (iv) from (iii), we get 9x+6y=33 (iii) 4x+6y=8 (iv) 5x =25 x=5
  • 14. • Putting the value of x in equation (ii) we get, 2x+3y=4 2x5+3y=4 10+3y=4 3y=4-10 3y=-6 y=-2