Points to Remember: Pair of Linear Equations in Two Variables

# Points to Remember: Pair of Linear Equations in Two Variables | Mathematics (Maths) Class 10 PDF Download

``` Page 1

A pair of linear
equation in two
Variables
Page 2

A pair of linear
equation in two
Variables
Substitution Method
Linear equation can be represented and solved by
Equation in the form of ax + by + c = 0, where a, b & c are real numbers and ‘a’ & ‘b’ both cannot be
zero is known as linear equation in two variables
Graphical Method Algebraic Method
Substitution Elimination Cross Multiplication
Cross multiplication method Elimination Method
.   We nd the value of one variable ( say y ) in
terms of the other variable ( say x ) from either of
the equations.
.   Substitute the value of y in another equation
to make it in one variable. Now we can get the
value of x.
.   Substitute the value of obtained variable x in
any of the equations to get the value of ‘y ’.
.   We make coecient of a variable in one of the
equations equal to the coecient of the same
variable in another equation by multiplying it with
a suitable number.
.   Add or subtract one equation from other to
eliminate one variable.
.   Now we get an equation in one variable and
can solve it to get its value.
.   Substitute the value of the obtained variable in
any of the equations to get the value of another
variable.
1
1 1
2
3
2
3
4
.   Let the general from of pair of linear
equations in two variables x & y be
a
1
x + b
1
y + c
1
= 0
a
2
x + b
2
y + c
2
= 0
2 .   x & y can be found by using following formulas
b
1
c
2
- b
2
c
1
a
1
b
2
- a
2
b
1
c
1
a
2
- c
2
a
1
a
1
b
2
- a
2
b
1
y =
x =
Methods to solve pair of linear equations
A PAIR OF LINEAR EQUATIONS IN TWO VARIABLES 7
A Pair of Linear Equations in Two Variables
{ Sotution }
Y
X
1
1
(0,0)
2 x + y = 6
x + y = 4 (2, 2)
x=2, y=2
2
2
3
3
4
4
5
5
6
6
7
7 8 9
.
Page 3

A pair of linear
equation in two
Variables
Substitution Method
Linear equation can be represented and solved by
Equation in the form of ax + by + c = 0, where a, b & c are real numbers and ‘a’ & ‘b’ both cannot be
zero is known as linear equation in two variables
Graphical Method Algebraic Method
Substitution Elimination Cross Multiplication
Cross multiplication method Elimination Method
.   We nd the value of one variable ( say y ) in
terms of the other variable ( say x ) from either of
the equations.
.   Substitute the value of y in another equation
to make it in one variable. Now we can get the
value of x.
.   Substitute the value of obtained variable x in
any of the equations to get the value of ‘y ’.
.   We make coecient of a variable in one of the
equations equal to the coecient of the same
variable in another equation by multiplying it with
a suitable number.
.   Add or subtract one equation from other to
eliminate one variable.
.   Now we get an equation in one variable and
can solve it to get its value.
.   Substitute the value of the obtained variable in
any of the equations to get the value of another
variable.
1
1 1
2
3
2
3
4
.   Let the general from of pair of linear
equations in two variables x & y be
a
1
x + b
1
y + c
1
= 0
a
2
x + b
2
y + c
2
= 0
2 .   x & y can be found by using following formulas
b
1
c
2
- b
2
c
1
a
1
b
2
- a
2
b
1
c
1
a
2
- c
2
a
1
a
1
b
2
- a
2
b
1
y =
x =
Methods to solve pair of linear equations
A PAIR OF LINEAR EQUATIONS IN TWO VARIABLES 7
A Pair of Linear Equations in Two Variables
{ Sotution }
Y
X
1
1
(0,0)
2 x + y = 6
x + y = 4 (2, 2)
x=2, y=2
2
2
3
3
4
4
5
5
6
6
7
7 8 9
.
A PAIR OF LINEAR EQUATIONS IN TWO VARIABLES 8
If
Condition of Consistency
Number of Solutions of a Pair of Linear Equations
Pair of Linear Equations
{ Lines are intersecting }
{ Unique solution }
{ Consistent }
Y
X
1
1
(0,0)
2 m + a = 6
m + a = 4
(2, 2)
2
2
3
3
4
4
5
5
6
6
7
7 8 9
.
{Lines are parallel}
{No solution}
{Inconsistent}
Y
X 1
1
(0,0)
m + a = 6
m + a = 4
2
2
3
3
4
4
5
5
6
6
7
7 8 9
= =
a
1
a
2
b
1
b
2
c
1
c
2
{Lines are coincident}
{Many solutions}
{Consistent}
Y
X
1
1
(0,0)
2
2
3
3
4
4
5
5
6
6
7
7 8 9
2m + 2a = 8
m + a = 4
= =
a
1
a
2
b
1
b
2
c
1
c
2
a
1
a
2
=
b
1
b
2
a
1
a
2
=
b
1
b
2
Number of Solutions Relation Between Lines
(Inconsistent) No Solution
(Inconsistent) No Solution
Coincident Lines
Parallel Lines
Parallel Lines
Intersecting Lines
Intersecting Lines
(Consistent) Unique Solution
(Consistent) Many Solution
(Consistent) Unique Solution
{ Then check for }
2x + 4y = 6 4x - 8y = 12 ;
x - 3y = 1 2x - 6y = 4 ;
2p - 3q = 2 3p - 2q = 2 ;
m + 4n = 1
3
2
3m + 8n = 2 ;
-6x + 2y = 3
-3x + y = 3
;
Page 4

A pair of linear
equation in two
Variables
Substitution Method
Linear equation can be represented and solved by
Equation in the form of ax + by + c = 0, where a, b & c are real numbers and ‘a’ & ‘b’ both cannot be
zero is known as linear equation in two variables
Graphical Method Algebraic Method
Substitution Elimination Cross Multiplication
Cross multiplication method Elimination Method
.   We nd the value of one variable ( say y ) in
terms of the other variable ( say x ) from either of
the equations.
.   Substitute the value of y in another equation
to make it in one variable. Now we can get the
value of x.
.   Substitute the value of obtained variable x in
any of the equations to get the value of ‘y ’.
.   We make coecient of a variable in one of the
equations equal to the coecient of the same
variable in another equation by multiplying it with
a suitable number.
.   Add or subtract one equation from other to
eliminate one variable.
.   Now we get an equation in one variable and
can solve it to get its value.
.   Substitute the value of the obtained variable in
any of the equations to get the value of another
variable.
1
1 1
2
3
2
3
4
.   Let the general from of pair of linear
equations in two variables x & y be
a
1
x + b
1
y + c
1
= 0
a
2
x + b
2
y + c
2
= 0
2 .   x & y can be found by using following formulas
b
1
c
2
- b
2
c
1
a
1
b
2
- a
2
b
1
c
1
a
2
- c
2
a
1
a
1
b
2
- a
2
b
1
y =
x =
Methods to solve pair of linear equations
A PAIR OF LINEAR EQUATIONS IN TWO VARIABLES 7
A Pair of Linear Equations in Two Variables
{ Sotution }
Y
X
1
1
(0,0)
2 x + y = 6
x + y = 4 (2, 2)
x=2, y=2
2
2
3
3
4
4
5
5
6
6
7
7 8 9
.
A PAIR OF LINEAR EQUATIONS IN TWO VARIABLES 8
If
Condition of Consistency
Number of Solutions of a Pair of Linear Equations
Pair of Linear Equations
{ Lines are intersecting }
{ Unique solution }
{ Consistent }
Y
X
1
1
(0,0)
2 m + a = 6
m + a = 4
(2, 2)
2
2
3
3
4
4
5
5
6
6
7
7 8 9
.
{Lines are parallel}
{No solution}
{Inconsistent}
Y
X 1
1
(0,0)
m + a = 6
m + a = 4
2
2
3
3
4
4
5
5
6
6
7
7 8 9
= =
a
1
a
2
b
1
b
2
c
1
c
2
{Lines are coincident}
{Many solutions}
{Consistent}
Y
X
1
1
(0,0)
2
2
3
3
4
4
5
5
6
6
7
7 8 9
2m + 2a = 8
m + a = 4
= =
a
1
a
2
b
1
b
2
c
1
c
2
a
1
a
2
=
b
1
b
2
a
1
a
2
=
b
1
b
2
Number of Solutions Relation Between Lines
(Inconsistent) No Solution
(Inconsistent) No Solution
Coincident Lines
Parallel Lines
Parallel Lines
Intersecting Lines
Intersecting Lines
(Consistent) Unique Solution
(Consistent) Many Solution
(Consistent) Unique Solution
{ Then check for }
2x + 4y = 6 4x - 8y = 12 ;
x - 3y = 1 2x - 6y = 4 ;
2p - 3q = 2 3p - 2q = 2 ;
m + 4n = 1
3
2
3m + 8n = 2 ;
-6x + 2y = 3
-3x + y = 3
;
Variable Reducible method
+ =
4
(x - y)
1
=  p =  q
(x - y)
1
(x + y)
3
(x + y)
2
Take
,
?
4p + 3q = 2
+ =
6
y
1
=  p
=  q
x
1
y
3
x
4
Take
Divide by x?
,
?
?
6x + 3y = 4 x?
6p + 3q = 4
?
?
+
=
2
m
7
Take
n
3
2x + 7y = 3
? = ,
1
m
1
n
x = y
Introduction to Pair of
Linear Equations
Problem Solving
of Interesting Lines
Conditions of Consistency
Scan the QR Codes to watch our free videos
?
?
?
?
Linear means straight and a linear equation refers to an equa-
tion which when plotted on the graph gives a straight line. And it is
true for any linear equation whether it is one variable, two variable or
multi-variable.
To solve linear equations, number of variables should be equal
to the number of equations.
If it is asked to solve linear equations and the method is not
mentioned then, one can use any of the methods which is most suita-
ble for that particular question.
There is a formula that shows the relation between time, speed
and distance. Use this formula for word problems.
Speed =
Distance
Time
There can be a variety of situations which can be mathematically represented by two equations which are not linear.
But we can eventually alter them and reduce them to a pair of linear equations.
A PAIR OF LINEAR EQUATIONS IN TWO VARIABLES 9
```

## Mathematics (Maths) Class 10

126 videos|477 docs|105 tests

## FAQs on Points to Remember: Pair of Linear Equations in Two Variables - Mathematics (Maths) Class 10

 1. What are the types of solutions for a pair of linear equations in two variables?
Ans. The types of solutions for a pair of linear equations in two variables are: 1. Unique Solution: When the pair of equations have a single common solution, it is called a unique solution. Graphically, it represents the point of intersection of the two lines. 2. No Solution: If the pair of equations represent parallel lines, they will never intersect. In this case, there is no common solution, and the system is said to have no solution. 3. Infinitely Many Solutions: When the two equations represent coincident lines, they will intersect at infinite points. In such cases, the system is said to have infinitely many solutions.
 2. How can we solve a pair of linear equations using the substitution method?
Ans. The substitution method is a technique to solve a pair of linear equations in two variables. Here are the steps to solve using the substitution method: 1. Choose one equation and solve it for one variable in terms of the other variable. 2. Substitute the expression obtained in step 1 into the other equation. 3. Solve the resulting equation to find the value of the remaining variable. 4. Substitute the value found in step 3 into any of the original equations to find the value of the other variable. 5. Verify the solution by substituting the values of both variables into both equations.
 3. What is the graphical representation of a pair of linear equations with no solution?
Ans. When a pair of linear equations has no solution, it means the equations represent parallel lines. Graphically, these parallel lines will never intersect. They will run side by side, maintaining the same distance between them. Thus, the graphical representation of a pair of linear equations with no solution is two parallel lines.
 4. How can we determine the number of solutions for a pair of linear equations using the determinant?
Ans. To determine the number of solutions for a pair of linear equations using the determinant, we use Cramer's rule. Here's how: 1. Write the given pair of equations in the form of: ax + by = c dx + ey = f 2. Calculate the determinants D, Dx, and Dy using the coefficients of the variables as follows: D = ae - bd Dx = ce - bf Dy = af - cd 3. If D ≠ 0, then the pair of equations has a unique solution given by: x = Dx / D y = Dy / D 4. If D = 0 and both Dx and Dy are nonzero, then the pair of equations has no solution. 5. If D = 0 and either Dx or Dy is zero, then the pair of equations has infinitely many solutions.
 5. Can a pair of linear equations have more than one unique solution?
Ans. No, a pair of linear equations cannot have more than one unique solution. By definition, a unique solution implies that there is only one point of intersection between the two lines represented by the equations. If there were multiple points of intersection, the lines would not be distinct, and the system would have infinitely many solutions.

## Mathematics (Maths) Class 10

126 videos|477 docs|105 tests

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