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# RD Sharma Solutions: Exercise 3.5 - Pair of Linear Equations in Two Variables Notes | EduRev

## Class 10 : RD Sharma Solutions: Exercise 3.5 - Pair of Linear Equations in Two Variables Notes | EduRev

``` Page 1

Exercise 3.5
In each of the following systems of equations determine whether the system has a unique
solution, no solution or infinitely many solutions. In case there is a unique solution, find it:
(1 -4)
1.
3 3 0
3 9 2 0
xy
xy
? ? ?
? ? ?

Sol:
The given system of equations may be written as
Page 2

Exercise 3.5
In each of the following systems of equations determine whether the system has a unique
solution, no solution or infinitely many solutions. In case there is a unique solution, find it:
(1 -4)
1.
3 3 0
3 9 2 0
xy
xy
? ? ?
? ? ?

Sol:
The given system of equations may be written as

3 3 0
3 9 2 0
xy
xy
? ? ?
? ? ?

The given system of equations is of the form
1 1 1
2 2 2
0
0
a x b y c
a x b y c
? ? ?
? ? ?

Where,
1 1 1
1, 3, 3 a b c ? ? ? ? ?
And
2 2 2
3, 9, 2 a b c ? ? ? ? ?
We have,
1
2
1
2
1
3
31
93
a
a
b
b
?
?
? ?
?

And,
1
2
33
22
c
c
?
? ?
?

Clearly,
1 1 1
2 2 2
a b c
a b c
??
So, the given system of equation has no solutions.

2.
2 5 0
4 2 10 0
xy
xy
? ? ?
? ? ?

Sol:
The given system of equation may be written as
2 5 0
4 2 10 0
xy
xy
? ? ?
? ? ?

The given system of equations is of the form
1 1 1
2 2 2
0
0
a x b y c
a x b y c
? ? ?
? ? ?

Where,
1 1 1
2, 1, 5 a b c ? ? ? ?
And
222
4, 2, 10 a b c ? ? ? ?
We have,
1
2
1
2
21
42
1
2
a
a
b
b
? ?
?

And,
1
2
51
10 2
c
c
?
? ?
?

Page 3

Exercise 3.5
In each of the following systems of equations determine whether the system has a unique
solution, no solution or infinitely many solutions. In case there is a unique solution, find it:
(1 -4)
1.
3 3 0
3 9 2 0
xy
xy
? ? ?
? ? ?

Sol:
The given system of equations may be written as

3 3 0
3 9 2 0
xy
xy
? ? ?
? ? ?

The given system of equations is of the form
1 1 1
2 2 2
0
0
a x b y c
a x b y c
? ? ?
? ? ?

Where,
1 1 1
1, 3, 3 a b c ? ? ? ? ?
And
2 2 2
3, 9, 2 a b c ? ? ? ? ?
We have,
1
2
1
2
1
3
31
93
a
a
b
b
?
?
? ?
?

And,
1
2
33
22
c
c
?
? ?
?

Clearly,
1 1 1
2 2 2
a b c
a b c
??
So, the given system of equation has no solutions.

2.
2 5 0
4 2 10 0
xy
xy
? ? ?
? ? ?

Sol:
The given system of equation may be written as
2 5 0
4 2 10 0
xy
xy
? ? ?
? ? ?

The given system of equations is of the form
1 1 1
2 2 2
0
0
a x b y c
a x b y c
? ? ?
? ? ?

Where,
1 1 1
2, 1, 5 a b c ? ? ? ?
And
222
4, 2, 10 a b c ? ? ? ?
We have,
1
2
1
2
21
42
1
2
a
a
b
b
? ?
?

And,
1
2
51
10 2
c
c
?
? ?
?

Clearly,
111
222
a b c
a b c
??
So, the given system of equation has infinity many solutions.

3.
3 5 20
6 10 40
xy
xy
? ?
? ?

Sol:
3 5 20
6 10 40
xy
xy
? ?
? ?

Compare it with
1 1 1
1 2 2
0
0
a x by c
a x by c
? ? ?
? ? ?

We get
1 3, 1 5 1 20
2 6, 2 10 2 40
a b and c
a b and c
? ? ? ? ?
? ? ? ? ?

1 1 1
2 2 2
3 5 20
,
6 10 40
a b c
and
a b c
? ?
? ? ?
??

Simplifying it we get
1 1 1
2 2 2
1 1 1
,
2 2 2
a b c
and
a b c
? ? ?
Hence
111
222
a b c
a b c
??
So both lines are coincident and overlap with each other
So, it will have infinite or many solutions

4.
2 8 0
5 10 10 0
xy
xy
? ? ?
? ? ?

Sol:
The given system of equation may be written as
2 8 0
5 10 10 0
xy
xy
? ? ?
? ? ?

The given system if equation is of the form
1 1 1
2 2 2
0
0
a x b y c
a x b y c
? ? ?
? ? ?

Where,
1 1 1
1, 2, 8 a b c ? ? ? ? ?
And,
2 2 2
5, 10, 10 a b c ? ? ? ? ?
Page 4

Exercise 3.5
In each of the following systems of equations determine whether the system has a unique
solution, no solution or infinitely many solutions. In case there is a unique solution, find it:
(1 -4)
1.
3 3 0
3 9 2 0
xy
xy
? ? ?
? ? ?

Sol:
The given system of equations may be written as

3 3 0
3 9 2 0
xy
xy
? ? ?
? ? ?

The given system of equations is of the form
1 1 1
2 2 2
0
0
a x b y c
a x b y c
? ? ?
? ? ?

Where,
1 1 1
1, 3, 3 a b c ? ? ? ? ?
And
2 2 2
3, 9, 2 a b c ? ? ? ? ?
We have,
1
2
1
2
1
3
31
93
a
a
b
b
?
?
? ?
?

And,
1
2
33
22
c
c
?
? ?
?

Clearly,
1 1 1
2 2 2
a b c
a b c
??
So, the given system of equation has no solutions.

2.
2 5 0
4 2 10 0
xy
xy
? ? ?
? ? ?

Sol:
The given system of equation may be written as
2 5 0
4 2 10 0
xy
xy
? ? ?
? ? ?

The given system of equations is of the form
1 1 1
2 2 2
0
0
a x b y c
a x b y c
? ? ?
? ? ?

Where,
1 1 1
2, 1, 5 a b c ? ? ? ?
And
222
4, 2, 10 a b c ? ? ? ?
We have,
1
2
1
2
21
42
1
2
a
a
b
b
? ?
?

And,
1
2
51
10 2
c
c
?
? ?
?

Clearly,
111
222
a b c
a b c
??
So, the given system of equation has infinity many solutions.

3.
3 5 20
6 10 40
xy
xy
? ?
? ?

Sol:
3 5 20
6 10 40
xy
xy
? ?
? ?

Compare it with
1 1 1
1 2 2
0
0
a x by c
a x by c
? ? ?
? ? ?

We get
1 3, 1 5 1 20
2 6, 2 10 2 40
a b and c
a b and c
? ? ? ? ?
? ? ? ? ?

1 1 1
2 2 2
3 5 20
,
6 10 40
a b c
and
a b c
? ?
? ? ?
??

Simplifying it we get
1 1 1
2 2 2
1 1 1
,
2 2 2
a b c
and
a b c
? ? ?
Hence
111
222
a b c
a b c
??
So both lines are coincident and overlap with each other
So, it will have infinite or many solutions

4.
2 8 0
5 10 10 0
xy
xy
? ? ?
? ? ?

Sol:
The given system of equation may be written as
2 8 0
5 10 10 0
xy
xy
? ? ?
? ? ?

The given system if equation is of the form
1 1 1
2 2 2
0
0
a x b y c
a x b y c
? ? ?
? ? ?

Where,
1 1 1
1, 2, 8 a b c ? ? ? ? ?
And,
2 2 2
5, 10, 10 a b c ? ? ? ? ?

We have,
1
2
1
2
1
5
21
10 5
a
a
b
b
?
?
??
?

And,
1
2
84
10 5
c
c
?
? ?
?

Clearly,
1 2 1
2 2 2
a b c
a b c
??
So, the given system of equation has no solution.

5.
2 5 0
3 1 0
kx y
xy
? ? ?
? ? ?

Sol:
The given system of equation is
2 5 0
3 1 0
kx y
xy
? ? ?
? ? ?

The system of equation is of the form
1 1 1
2 2 2
0
0
a x b y c
a x b y c
? ? ?
? ? ?

Where,
1 1 1
, 2, 5 a k b c ? ? ? ?
And,
2 2 2
3, 1, 1 a b c ? ? ? ?
For a unique solution, we must have
11
22
2
31
6
ab
ab
k
k
?
? ?
??

So, the given system of equations will have a unique solution for all real values of k other
than 6.

6. 4x + ky + 8 = 0
2x + 2y + 2 = 0
Sol:
Here
1 2 1 2
4, , 2, 2 a a k b b ? ? ? ?
Now for the given pair to have a unique solution:
11
22
ab
ab
?
Page 5

Exercise 3.5
In each of the following systems of equations determine whether the system has a unique
solution, no solution or infinitely many solutions. In case there is a unique solution, find it:
(1 -4)
1.
3 3 0
3 9 2 0
xy
xy
? ? ?
? ? ?

Sol:
The given system of equations may be written as

3 3 0
3 9 2 0
xy
xy
? ? ?
? ? ?

The given system of equations is of the form
1 1 1
2 2 2
0
0
a x b y c
a x b y c
? ? ?
? ? ?

Where,
1 1 1
1, 3, 3 a b c ? ? ? ? ?
And
2 2 2
3, 9, 2 a b c ? ? ? ? ?
We have,
1
2
1
2
1
3
31
93
a
a
b
b
?
?
? ?
?

And,
1
2
33
22
c
c
?
? ?
?

Clearly,
1 1 1
2 2 2
a b c
a b c
??
So, the given system of equation has no solutions.

2.
2 5 0
4 2 10 0
xy
xy
? ? ?
? ? ?

Sol:
The given system of equation may be written as
2 5 0
4 2 10 0
xy
xy
? ? ?
? ? ?

The given system of equations is of the form
1 1 1
2 2 2
0
0
a x b y c
a x b y c
? ? ?
? ? ?

Where,
1 1 1
2, 1, 5 a b c ? ? ? ?
And
222
4, 2, 10 a b c ? ? ? ?
We have,
1
2
1
2
21
42
1
2
a
a
b
b
? ?
?

And,
1
2
51
10 2
c
c
?
? ?
?

Clearly,
111
222
a b c
a b c
??
So, the given system of equation has infinity many solutions.

3.
3 5 20
6 10 40
xy
xy
? ?
? ?

Sol:
3 5 20
6 10 40
xy
xy
? ?
? ?

Compare it with
1 1 1
1 2 2
0
0
a x by c
a x by c
? ? ?
? ? ?

We get
1 3, 1 5 1 20
2 6, 2 10 2 40
a b and c
a b and c
? ? ? ? ?
? ? ? ? ?

1 1 1
2 2 2
3 5 20
,
6 10 40
a b c
and
a b c
? ?
? ? ?
??

Simplifying it we get
1 1 1
2 2 2
1 1 1
,
2 2 2
a b c
and
a b c
? ? ?
Hence
111
222
a b c
a b c
??
So both lines are coincident and overlap with each other
So, it will have infinite or many solutions

4.
2 8 0
5 10 10 0
xy
xy
? ? ?
? ? ?

Sol:
The given system of equation may be written as
2 8 0
5 10 10 0
xy
xy
? ? ?
? ? ?

The given system if equation is of the form
1 1 1
2 2 2
0
0
a x b y c
a x b y c
? ? ?
? ? ?

Where,
1 1 1
1, 2, 8 a b c ? ? ? ? ?
And,
2 2 2
5, 10, 10 a b c ? ? ? ? ?

We have,
1
2
1
2
1
5
21
10 5
a
a
b
b
?
?
??
?

And,
1
2
84
10 5
c
c
?
? ?
?

Clearly,
1 2 1
2 2 2
a b c
a b c
??
So, the given system of equation has no solution.

5.
2 5 0
3 1 0
kx y
xy
? ? ?
? ? ?

Sol:
The given system of equation is
2 5 0
3 1 0
kx y
xy
? ? ?
? ? ?

The system of equation is of the form
1 1 1
2 2 2
0
0
a x b y c
a x b y c
? ? ?
? ? ?

Where,
1 1 1
, 2, 5 a k b c ? ? ? ?
And,
2 2 2
3, 1, 1 a b c ? ? ? ?
For a unique solution, we must have
11
22
2
31
6
ab
ab
k
k
?
? ?
??

So, the given system of equations will have a unique solution for all real values of k other
than 6.

6. 4x + ky + 8 = 0
2x + 2y + 2 = 0
Sol:
Here
1 2 1 2
4, , 2, 2 a a k b b ? ? ? ?
Now for the given pair to have a unique solution:
11
22
ab
ab
?

i.e.,
4
22
k
?
i.e., 4 k ?
Therefore, for all values of k, except 4, the given pair of equations will have a unique
solution.

7.
45
2 3 12
x y k
xy
??
? ?

Sol:
The given system of equation is
4 5 0
2 3 12 0
x y k
xy
? ? ?
? ? ?

The system of equation is of the form
1 1 1
2 2 2
0
0
a x b y c
a x b y c
? ? ?
? ? ?

Where,
1 1 1
4, 5, a b c k ? ? ? ? ?
And,
2 2 2
2, 3, 12 a b c ? ? ? ? ?
For a unique solution, we must have
11
22
45
23
ab
ab
?
?
? ?
?

k ? is any real number.
So, the given system of equations will have a unique solution for all real values of k.

8.
23
5 7 0
xy
x ky
? ?
? ? ?

Sol:
The given system of equation is
2 3 0
5 7 0
xy
x ky
? ? ?
? ? ?

The system of equation is of the form
1 1 1
2 2 2
0
0
a x b y c
a x b y c
? ? ?
? ? ?

Where,
1 1 1
1, 2, 3 a b c ? ? ? ?
And,
2 2 2
5, , 7 a b k c ? ? ?
For a unique solution, we must have
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## Mathematics (Maths) Class 10

62 videos|363 docs|103 tests

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