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Linear Equations in
Two Variables
MADE BY :
1. Shekhar
2. Suhail khan
3. Hansika
4. Nihal
5. Akash kr.
6. Babita
7. Kanchan
IXth –C
Equations of the form ax + by = c are called linear equations in two
variables.Where A, B, and C are real numbers and A and B are not bothzero.
The point (0,4) is the y-intercept.
The point (6,0) is the x-intercept.
x
y
2-2
This is the graph of the equation
2x + 3y = 12.
(0,4)
(6,0)
The graph of any linear equation in two variables is a straight line
Equations of the form
𝒂𝟏
𝒃𝟐
≠
𝒃𝟏
𝒃𝟐
are called INTERSECTING LINES
These type of equations have Exactly one solution(unique)
These are consistent lines
Example: x-2y=0
3x+4y=20
𝑎1
𝑎2
=
1
3
𝑏1
𝑏2
=
−2
4
𝑎1
𝑎2
≠
𝑏1
𝑏2
Equations of the form
𝒂𝟏
𝒃𝟐
=
𝒃𝟏
𝒃𝟐
=
𝒄𝟏
𝒄𝟐
are called COINCIDENT LINES
These type of equations have Infinity many no. of solution.
These are consistent lines
Example: 2x+3y=9
4x+6y=18
𝑎1
𝑎2
=
1
2
𝑏1
𝑏2
=
1
2
𝑎1
𝑎2
=
𝑏1
𝑏2
=
𝑐1
𝑐2
𝑐1
𝑐2
=
1
2
Equations of the form
𝒂𝟏
𝒃𝟐
=
𝒃𝟏
𝒃𝟐
≠
𝒄𝟏
𝒄𝟐
are called PARALLEL LINES
These type of equations have no solution.
These are inconsistent lines
Example: x+2y=4
2x+4y=12
𝑎1
𝑎2
=
𝑏1
𝑏2
≠
𝑐1
𝑐2
𝑎1
𝑎2
=
1
2
𝑏1
𝑏2
=
1
2
𝑐1
𝑐2
=
1
3
EXAMPLE :
X + y=10
X - y=4
x 5 6 7
y 5 4 3
x 3 2 5
y -1 -2 1
Example: x + y=5
2x+2y=10
x 0 2 1
y 5 3 4
Example: 2x + y=160
2x+y=150
x 60 70 50
y 40 20 60
x 60 50 40
y 30 50 70
EXAMPLE : x + y=14
x – y=14
Solution : x + y=14-------------------------1
x – y=14-------------------------2
From equation 1
x =14-y
Put the value of x in equation 2
14-y-y=4
14-2y=4
14-4=2y
10=2y
Y=5
x =14-5
X=9
EXAMPLE : x + y=25
x + 2y=40
x + y=25-------------------------1
x + 2y=40-------------------------2
Solution :
Subtract equation 2 from 1
x + 2y=40
- - -
x + y=25
-y=-15
-y=-15
Y=15
x + 15=25
X=25-15
X=10
The general form of cross- multiplication method is:
𝒙
𝒃𝟏𝒄𝟐−𝒃𝟐𝒄𝟏
=
𝒚
𝒄𝟏𝒂𝟐−𝒄𝟐𝒂𝟏
=
𝟏
𝒂𝟏𝒃𝟐−𝒂𝟐𝒃𝟏
LINEAR EQUATION SYSTEM
with 2 variables
DO YOU REMEMBER?
 Which one of the following example is linear
equation of one variable? Give your reason.
2
1
a) (4 2) 3
2
b) 3 2 5 0
c) 3 2 7 2
d) 2 3
x
x x
y y
x y
 
  
  
 
The point (a) and (c)
are examples of linear
equation of one
variable.
Can you find the
solution?
 If I have
what the meaning of solution of that
system?
 In how many ways we can solve linear
equation system?
we can solve linear equation system in
four ways, that are substitution,
elimination, substitution-elimination and
graph method
2 4
2 3 12
x y
x y
 
 
LINEAR EQUATION SYTEM WITH 2 VARIABLES
 In general, a linear equation system of
2 variables x and y can be expressed:
 Let’s try to solve this problem
1 1 1
2 2 2
a x b y c
a x b y c
 
  1 1 1 2 2 2where , , , , , anda b c a b c R
2 4
2 3 12
x y
x y
 
 
 ( , )SS x y
LINEAR EQUATION SYTEM WITH 2 VARIABLES
 Graphic Method
1. Draw a line 2x – y
= 4. What is the
solution of that
linear equation?
2. Also draw a line
2x + 3y = 12. What
is the solution of
that linear
equation?
LINEAR EQUATION SYTEM WITH 2 VARIABLES
 So, can you guess the solution of that
linear equation system?
 Conclusion:
The solution of that linear system is the
point of intersection of the lines
 There are 4 ways to solve linear equation
system with 2 variables :
 Substitution
 Elimination
 Elimination-substitution
 graph method
- Using Substitution Method –
1. Write one of the equation in the form
2. Substitute y (or x) obtained in the
first step into the other equations
3. Solve the equations to obtain the value
4. Substitute the value x=x1 obtained to
get y1 or substitute the value y=y1
obtained to get x1
5. The solution set is
ory ax b x cy d   
1 1orx x y y 
  1 1,x y
 Using Elimination Method –
 The procedure for eliminating variable x (or
y)
1. Consider the coefficient of x (or y). If they
have same sign, subtract the equation (1) from
equation (2), if they have different sign add
them.
2. If the coefficients are different, make them same
by multiplying each of the equations with the
corresponding constants, then do the addition or
subtraction as the first step.
1 1 1
2 2 2
............. (1)
............. (2)
a x b y c
a x b y c
 
 
 Using Graphing Method –
The solution of the linear system with two
variables is the point of intersection of the
lines.
 If 2 lines are drawn in the same coordinate
there are 3 possibilities of solution :
 The lines will intersect at exactly one point if the
gradients of the lines are different. Then the
solution is unique
 The lines are parallel if the gardients of the line
are same. Thus, there are no solution
 The line are coincide to each other if one line is a
multiple of each other. Thus the solution are
infinite
SOLVING LINEAR EQUATIONS
1-STEP EQUATIONS: + AND –
In elementary school, you solved
equations that looked like this:
In Algebra, we just use a variable
instead of the box!
We could rewrite the same
equation like this: x + 3 = 5
TO SOLVE:
Need to get x by itself.
We must “undo” whatever has been
done to the variable.
Do the same to BOTH SIDES of the =
sign!
Answer should be: x = some number.
EQUATIONS HAVE TO BALANCE!
Both sides of the = sign must “weigh”
the same!
When you change one side, do the
same to the other side, so it stays
balanced!
x + 3 = 7
EXAMPLES:
Solve each equation. Check your
answers!
x + 3 = -7 m – 6 = 8
4 + a = -3
-8 + y = 7
24 = 12 + t
-17 = x – 2
YOU TRY!
Solve each equation. Check your
answers!
x – 9 = -6 16 + n = 10
1-STEP EQUATIONS: × AND ÷
Just like with adding and subtracting, we
need to “undo” what is done to x.
Use division to “undo” multiplication.
Use multiplication to “undo” division.
Remember: A fraction bar means divide!
Example: says x divided by 3!
EXAMPLES:
Solve each equation:
2x = 8
12 = -3b
YOU TRY!
Solve each 1-step equation.
-4a = 16
BASIC 2-STEP EQUATIONS
A lot like 1-step equations!
Goal: Get x by itself!
Need to “undo” what is done to x.
Always do the same thing to BOTH
SIDES!
Use Order of Operations backwards:
 Add/Subtract first
 Multiply/Divide last
EXAMPLES:
4x + 7 = 27 – 6 – 2y = 10
5 = 7r - 2 -4 = 2 - m
YOU TRY!
Solve each equation. Check your
answers!
-3x + 7 = 16 -8 = 2a + 2
MULTI-STEP EQUATIONS
Multi-step equations have ( ).
To solve:
 Distribute first!!
 Then, solve like a 2-step equation!
Example: 4(-x + 4) = 12
EXAMPLES:
3(– 6 + 3x) = 18
30 = -5(6n + 6)
10 (1 + 3b) = -20
-2 = -(n – 8)
YOUR TRY!
-6(1 – 5v) = 54
2 VARIABLES- SAME SIDE
Some equations have variables in
more than one place.
When variables are on the same
side of the = sign:
 First, Combine Like Terms
Like terms may NOT cross the = sign!
 Then, solve like a 2-step
EXAMPLES:
2x + 3x + 1 = 36
3 + 6x – 2x = -5
-26 = -7x + 6 + 3x
YOU TRY!
Solve for x.
8x – 3x + 7 = 17
VARIABLES ON BOTH SIDES
To solve equations like this:
2x – 4 = 2 + 5x
1. Get variables together on the
left.
Add or subtract right side x term
from both sides.
2. Solve like a 2-step equation.
EXAMPLES
9 – 7x = -5x – 7
-3x – 7 = 2x – 13
YOU TRY!
-3x + 9 = 44 + 4x
UNDERSTANDING BASIC TERMS
LINEAR EQUATION- Straight line or degree 1
EQUATION- Any mathematical expression
equating to 0.
VARIABLE- Any alphabetical constants are
called variable.
Linear Equation
In one variable In two variable
SOLUTION OF LINEAR EQUATION
Solution of Linear
Equation
Graphical Method Algebric Method
REPRESENTATION OF LINEAR EQUATION
IN TWO VARIABLES
Representation of linear
Equation in two variable
In algebric form In word Problem
X in terms of y Y in terms of x
No of solution in a linear Equation
In one Variable
It contains only one solution for
Variable.
In two variables
It contain many or infinite number of
solution for variables.
Maths

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Maths

  • 1. Linear Equations in Two Variables MADE BY : 1. Shekhar 2. Suhail khan 3. Hansika 4. Nihal 5. Akash kr. 6. Babita 7. Kanchan IXth –C
  • 2. Equations of the form ax + by = c are called linear equations in two variables.Where A, B, and C are real numbers and A and B are not bothzero. The point (0,4) is the y-intercept. The point (6,0) is the x-intercept. x y 2-2 This is the graph of the equation 2x + 3y = 12. (0,4) (6,0) The graph of any linear equation in two variables is a straight line
  • 3. Equations of the form 𝒂𝟏 𝒃𝟐 ≠ 𝒃𝟏 𝒃𝟐 are called INTERSECTING LINES These type of equations have Exactly one solution(unique) These are consistent lines Example: x-2y=0 3x+4y=20 𝑎1 𝑎2 = 1 3 𝑏1 𝑏2 = −2 4 𝑎1 𝑎2 ≠ 𝑏1 𝑏2
  • 4. Equations of the form 𝒂𝟏 𝒃𝟐 = 𝒃𝟏 𝒃𝟐 = 𝒄𝟏 𝒄𝟐 are called COINCIDENT LINES These type of equations have Infinity many no. of solution. These are consistent lines Example: 2x+3y=9 4x+6y=18 𝑎1 𝑎2 = 1 2 𝑏1 𝑏2 = 1 2 𝑎1 𝑎2 = 𝑏1 𝑏2 = 𝑐1 𝑐2 𝑐1 𝑐2 = 1 2
  • 5. Equations of the form 𝒂𝟏 𝒃𝟐 = 𝒃𝟏 𝒃𝟐 ≠ 𝒄𝟏 𝒄𝟐 are called PARALLEL LINES These type of equations have no solution. These are inconsistent lines Example: x+2y=4 2x+4y=12 𝑎1 𝑎2 = 𝑏1 𝑏2 ≠ 𝑐1 𝑐2 𝑎1 𝑎2 = 1 2 𝑏1 𝑏2 = 1 2 𝑐1 𝑐2 = 1 3
  • 6. EXAMPLE : X + y=10 X - y=4 x 5 6 7 y 5 4 3 x 3 2 5 y -1 -2 1
  • 7. Example: x + y=5 2x+2y=10 x 0 2 1 y 5 3 4
  • 8. Example: 2x + y=160 2x+y=150 x 60 70 50 y 40 20 60 x 60 50 40 y 30 50 70
  • 9. EXAMPLE : x + y=14 x – y=14 Solution : x + y=14-------------------------1 x – y=14-------------------------2 From equation 1 x =14-y Put the value of x in equation 2 14-y-y=4 14-2y=4 14-4=2y 10=2y Y=5 x =14-5 X=9
  • 10. EXAMPLE : x + y=25 x + 2y=40 x + y=25-------------------------1 x + 2y=40-------------------------2 Solution : Subtract equation 2 from 1 x + 2y=40 - - - x + y=25 -y=-15 -y=-15 Y=15 x + 15=25 X=25-15 X=10
  • 11. The general form of cross- multiplication method is: 𝒙 𝒃𝟏𝒄𝟐−𝒃𝟐𝒄𝟏 = 𝒚 𝒄𝟏𝒂𝟐−𝒄𝟐𝒂𝟏 = 𝟏 𝒂𝟏𝒃𝟐−𝒂𝟐𝒃𝟏
  • 13. DO YOU REMEMBER?  Which one of the following example is linear equation of one variable? Give your reason. 2 1 a) (4 2) 3 2 b) 3 2 5 0 c) 3 2 7 2 d) 2 3 x x x y y x y           The point (a) and (c) are examples of linear equation of one variable. Can you find the solution?
  • 14.  If I have what the meaning of solution of that system?  In how many ways we can solve linear equation system? we can solve linear equation system in four ways, that are substitution, elimination, substitution-elimination and graph method 2 4 2 3 12 x y x y    
  • 15. LINEAR EQUATION SYTEM WITH 2 VARIABLES  In general, a linear equation system of 2 variables x and y can be expressed:  Let’s try to solve this problem 1 1 1 2 2 2 a x b y c a x b y c     1 1 1 2 2 2where , , , , , anda b c a b c R 2 4 2 3 12 x y x y      ( , )SS x y
  • 16. LINEAR EQUATION SYTEM WITH 2 VARIABLES  Graphic Method 1. Draw a line 2x – y = 4. What is the solution of that linear equation? 2. Also draw a line 2x + 3y = 12. What is the solution of that linear equation?
  • 17. LINEAR EQUATION SYTEM WITH 2 VARIABLES  So, can you guess the solution of that linear equation system?  Conclusion: The solution of that linear system is the point of intersection of the lines
  • 18.  There are 4 ways to solve linear equation system with 2 variables :  Substitution  Elimination  Elimination-substitution  graph method
  • 19. - Using Substitution Method – 1. Write one of the equation in the form 2. Substitute y (or x) obtained in the first step into the other equations 3. Solve the equations to obtain the value 4. Substitute the value x=x1 obtained to get y1 or substitute the value y=y1 obtained to get x1 5. The solution set is ory ax b x cy d    1 1orx x y y    1 1,x y
  • 20.  Using Elimination Method –  The procedure for eliminating variable x (or y) 1. Consider the coefficient of x (or y). If they have same sign, subtract the equation (1) from equation (2), if they have different sign add them. 2. If the coefficients are different, make them same by multiplying each of the equations with the corresponding constants, then do the addition or subtraction as the first step. 1 1 1 2 2 2 ............. (1) ............. (2) a x b y c a x b y c    
  • 21.  Using Graphing Method – The solution of the linear system with two variables is the point of intersection of the lines.  If 2 lines are drawn in the same coordinate there are 3 possibilities of solution :  The lines will intersect at exactly one point if the gradients of the lines are different. Then the solution is unique  The lines are parallel if the gardients of the line are same. Thus, there are no solution  The line are coincide to each other if one line is a multiple of each other. Thus the solution are infinite
  • 23. 1-STEP EQUATIONS: + AND – In elementary school, you solved equations that looked like this: In Algebra, we just use a variable instead of the box! We could rewrite the same equation like this: x + 3 = 5
  • 24. TO SOLVE: Need to get x by itself. We must “undo” whatever has been done to the variable. Do the same to BOTH SIDES of the = sign! Answer should be: x = some number.
  • 25. EQUATIONS HAVE TO BALANCE! Both sides of the = sign must “weigh” the same! When you change one side, do the same to the other side, so it stays balanced! x + 3 = 7
  • 26. EXAMPLES: Solve each equation. Check your answers! x + 3 = -7 m – 6 = 8
  • 27. 4 + a = -3 -8 + y = 7
  • 28. 24 = 12 + t -17 = x – 2
  • 29. YOU TRY! Solve each equation. Check your answers! x – 9 = -6 16 + n = 10
  • 30. 1-STEP EQUATIONS: × AND ÷ Just like with adding and subtracting, we need to “undo” what is done to x. Use division to “undo” multiplication. Use multiplication to “undo” division. Remember: A fraction bar means divide! Example: says x divided by 3!
  • 32.
  • 33. YOU TRY! Solve each 1-step equation. -4a = 16
  • 34. BASIC 2-STEP EQUATIONS A lot like 1-step equations! Goal: Get x by itself! Need to “undo” what is done to x. Always do the same thing to BOTH SIDES! Use Order of Operations backwards:  Add/Subtract first  Multiply/Divide last
  • 35. EXAMPLES: 4x + 7 = 27 – 6 – 2y = 10
  • 36. 5 = 7r - 2 -4 = 2 - m
  • 37. YOU TRY! Solve each equation. Check your answers! -3x + 7 = 16 -8 = 2a + 2
  • 38. MULTI-STEP EQUATIONS Multi-step equations have ( ). To solve:  Distribute first!!  Then, solve like a 2-step equation! Example: 4(-x + 4) = 12
  • 39. EXAMPLES: 3(– 6 + 3x) = 18 30 = -5(6n + 6)
  • 40. 10 (1 + 3b) = -20 -2 = -(n – 8)
  • 42. 2 VARIABLES- SAME SIDE Some equations have variables in more than one place. When variables are on the same side of the = sign:  First, Combine Like Terms Like terms may NOT cross the = sign!  Then, solve like a 2-step
  • 44. 3 + 6x – 2x = -5
  • 45. -26 = -7x + 6 + 3x
  • 46. YOU TRY! Solve for x. 8x – 3x + 7 = 17
  • 47. VARIABLES ON BOTH SIDES To solve equations like this: 2x – 4 = 2 + 5x 1. Get variables together on the left. Add or subtract right side x term from both sides. 2. Solve like a 2-step equation.
  • 48. EXAMPLES 9 – 7x = -5x – 7
  • 49. -3x – 7 = 2x – 13
  • 50. YOU TRY! -3x + 9 = 44 + 4x
  • 51.
  • 52. UNDERSTANDING BASIC TERMS LINEAR EQUATION- Straight line or degree 1 EQUATION- Any mathematical expression equating to 0. VARIABLE- Any alphabetical constants are called variable.
  • 53. Linear Equation In one variable In two variable
  • 54. SOLUTION OF LINEAR EQUATION Solution of Linear Equation Graphical Method Algebric Method
  • 55. REPRESENTATION OF LINEAR EQUATION IN TWO VARIABLES Representation of linear Equation in two variable In algebric form In word Problem X in terms of y Y in terms of x
  • 56. No of solution in a linear Equation In one Variable It contains only one solution for Variable. In two variables It contain many or infinite number of solution for variables.