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Submitted By:-
Deepak Saxena of Class xth
‘B’
Session:2011-2012
People’s Public School
“ The principal use of the analytic art is to bring mathematical
problem to equations and to exhibit those equations in the
most simple terms that can be .”
Contents :-
• Introduction
• Linear equations
• Points for solving a linear equation
• Solution of a linear equation
• Graph of a linear equation in two variables
• Equations of lines parallel to x-axis and y-axis
• Examples and solutions
• summary
Introduction
A simple linear equation is an
equality between two algebraic
expressions involving an unknown
value called the variable. In a
linear equation the exponent of
the variable is always equal to 1.
The two sides of an equation
are called Right Hand Side
(RHS) and Left-Hand Side
(LHS). They are written on
either side of equal sign.
Equation Lhs Rhs
4x + 3 = 5 4x + 3 5
2x + 5y = 0 2x + 5y 0
-2x + 3y = 6 -2x + 3y 6
Cont…
A linear equation in two variables
can be written in the form of ax +
by = c, where a, b, c are real
numbers, and a, b are not equal to
zero.
Equation a b c
2x+3y=9 2 3 -9
X+y/4-4=0 1 1/4 -4
5=2x 2 0 5
Y-2=0 0 1 -2
2+x/3=0 1/3 0 2
Linear equation :-
A linear equation is an algebraic equation in which each term is
either a constant or the product of a constant and a single
variable.
Linear equations can have one or more variables. Linear equations
occur with great regularity in applied mathematics. While they
arise quite naturally when modeling many phenomena, they are
particularly useful since many non-linear equations may be
reduced to linear equations by assuming that quantities of
interest vary to only a small extent from some "background"
state
5X+2=0 -2/5
-5 -4 -3 -2 -1 0 1 2 3 4 5
Solution of a linear equation
Every linear equation has a
unique solution as there is a
single variable in the equation
to be solved but in a linear
equation involving two variables
in the equation, a solution
means a pair of values, one for x
and one for y which satisfy the
given equation
Example-p (x)=2x+3y
(1)If x=3
2x + 3y = (2x3) + (3xy) = 12
6 + 3y = 12
y = 2,
therefore the solution is (3,2)
(2)If x = 2
2x + 3y = (2x2) + (3xy) = 12
4 + 3y = 12
Y = 8/3,
therefore the solution is
(2,8/3)
Similarly many another solutions
can be taken out from this
single equation. That is ,a linear
equation in two variables has
infinitely many solutions.
Graph of a linear equation in two variables
Graph of a linear equation is
representation of the linear
equation geo.
Observations on a graph :-
Every point whose coordinates
satisfy the equation lies on the
line AB.
Every point on the line AB gives a
solution of the equation.
Any point, which does not lie on the
line AB is not a solution of equation.
X+2Y=6
Equations of lines parallel to x-axis
The graph of y=a is a straight
line parallel to the x-axis
2y-7=1
2y-7+7=1+7
2y=8
2y/2=8/2
y=4
y=4
x
y
(2y-7=1)
Equations of lines parallel to y-axis
The graph of x=a is a straight line
parallel to the y-axis
x
3x-10=5
3x=15
x=5
x=5
(3x-10=5)
Examples and solutions
Give the values of a, b and c :
1) -2x+3y=9
a=-2 b=3 c=-9
2) 5x-3y=-4
a=5 b=-3 c=4
3) 3x+2=0
a=3 b=0 c=2
4) Y-5=0
a=0 b=1 c=-5
Write 2 solutions for each:
1) X+2y=6
If y=1;x=4
If y=2;x=2
2) 2x+y=4
If x=1;y=2
If x=2;y=0
3) 4x-2y=6
If x=1;y=-1
If x=2;y=1
Examples and solutions
Draw the graph of the equation:
2+2y=6x
If x=2;y=5
If x=1;y=2
If x=0;y=-1
(1,2)
(0,-1)
(2,5)
2+2y=6x
Examples and solutions
Examples and solutions
Give the geometric representation
of 2x+8=0 as an equation in two
variables:
y=-4
(2x+8=0)
(-4,3)(-4,-3)
y=-4
(2x+8=0)
(-4,3)(-4,-3)
SUMMARY
1) An equation of the form
ax +by + c =0,wherea,b and c
are real numbers, such that a and
b are not both zero, is called a
linear equation in two variables.
2) A linear equation in two
variables has infinitely many
solutions.
3) The graph of every linear
equation in two variables is a
straight line.
4) X=0 is the equation of the y-
axis and y=0 is the equation of
the x-axis
5) The graph of x=a is a straight
line parallel to the y-axis.
6) The graph of y=a is a straight
line parallel to the x-axis.
7) An equation of the type y=mx
represents a line passing through
the origin.
8) Every point on the graph of a
linear equation in two variables is
a solution of the linear equation.
Moreover, every solution of the
linear equation is a point on the
graph of the linear equation.
THANKS
FOR BEING
PATIENT

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linear equation

  • 1. Submitted By:- Deepak Saxena of Class xth ‘B’ Session:2011-2012 People’s Public School
  • 2. “ The principal use of the analytic art is to bring mathematical problem to equations and to exhibit those equations in the most simple terms that can be .”
  • 3. Contents :- • Introduction • Linear equations • Points for solving a linear equation • Solution of a linear equation • Graph of a linear equation in two variables • Equations of lines parallel to x-axis and y-axis • Examples and solutions • summary
  • 4. Introduction A simple linear equation is an equality between two algebraic expressions involving an unknown value called the variable. In a linear equation the exponent of the variable is always equal to 1. The two sides of an equation are called Right Hand Side (RHS) and Left-Hand Side (LHS). They are written on either side of equal sign. Equation Lhs Rhs 4x + 3 = 5 4x + 3 5 2x + 5y = 0 2x + 5y 0 -2x + 3y = 6 -2x + 3y 6
  • 5. Cont… A linear equation in two variables can be written in the form of ax + by = c, where a, b, c are real numbers, and a, b are not equal to zero. Equation a b c 2x+3y=9 2 3 -9 X+y/4-4=0 1 1/4 -4 5=2x 2 0 5 Y-2=0 0 1 -2 2+x/3=0 1/3 0 2
  • 6. Linear equation :- A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. Linear equations can have one or more variables. Linear equations occur with great regularity in applied mathematics. While they arise quite naturally when modeling many phenomena, they are particularly useful since many non-linear equations may be reduced to linear equations by assuming that quantities of interest vary to only a small extent from some "background" state 5X+2=0 -2/5 -5 -4 -3 -2 -1 0 1 2 3 4 5
  • 7. Solution of a linear equation Every linear equation has a unique solution as there is a single variable in the equation to be solved but in a linear equation involving two variables in the equation, a solution means a pair of values, one for x and one for y which satisfy the given equation Example-p (x)=2x+3y (1)If x=3 2x + 3y = (2x3) + (3xy) = 12 6 + 3y = 12 y = 2, therefore the solution is (3,2) (2)If x = 2 2x + 3y = (2x2) + (3xy) = 12 4 + 3y = 12 Y = 8/3, therefore the solution is (2,8/3) Similarly many another solutions can be taken out from this single equation. That is ,a linear equation in two variables has infinitely many solutions.
  • 8. Graph of a linear equation in two variables Graph of a linear equation is representation of the linear equation geo. Observations on a graph :- Every point whose coordinates satisfy the equation lies on the line AB. Every point on the line AB gives a solution of the equation. Any point, which does not lie on the line AB is not a solution of equation. X+2Y=6
  • 9. Equations of lines parallel to x-axis The graph of y=a is a straight line parallel to the x-axis 2y-7=1 2y-7+7=1+7 2y=8 2y/2=8/2 y=4 y=4 x y (2y-7=1)
  • 10. Equations of lines parallel to y-axis The graph of x=a is a straight line parallel to the y-axis x 3x-10=5 3x=15 x=5 x=5 (3x-10=5)
  • 11. Examples and solutions Give the values of a, b and c : 1) -2x+3y=9 a=-2 b=3 c=-9 2) 5x-3y=-4 a=5 b=-3 c=4 3) 3x+2=0 a=3 b=0 c=2 4) Y-5=0 a=0 b=1 c=-5
  • 12. Write 2 solutions for each: 1) X+2y=6 If y=1;x=4 If y=2;x=2 2) 2x+y=4 If x=1;y=2 If x=2;y=0 3) 4x-2y=6 If x=1;y=-1 If x=2;y=1 Examples and solutions
  • 13. Draw the graph of the equation: 2+2y=6x If x=2;y=5 If x=1;y=2 If x=0;y=-1 (1,2) (0,-1) (2,5) 2+2y=6x Examples and solutions
  • 14. Examples and solutions Give the geometric representation of 2x+8=0 as an equation in two variables: y=-4 (2x+8=0) (-4,3)(-4,-3) y=-4 (2x+8=0) (-4,3)(-4,-3)
  • 15. SUMMARY 1) An equation of the form ax +by + c =0,wherea,b and c are real numbers, such that a and b are not both zero, is called a linear equation in two variables. 2) A linear equation in two variables has infinitely many solutions. 3) The graph of every linear equation in two variables is a straight line. 4) X=0 is the equation of the y- axis and y=0 is the equation of the x-axis 5) The graph of x=a is a straight line parallel to the y-axis. 6) The graph of y=a is a straight line parallel to the x-axis. 7) An equation of the type y=mx represents a line passing through the origin. 8) Every point on the graph of a linear equation in two variables is a solution of the linear equation. Moreover, every solution of the linear equation is a point on the graph of the linear equation.