RD Sharma Solutions (Part 3): Pair of Linear Equations in Two Variables Class 10 Notes | EduRev

Mathematics (Maths) Class 10

Class 10 : RD Sharma Solutions (Part 3): Pair of Linear Equations in Two Variables Class 10 Notes | EduRev

 Page 1


 
 
 
 
 
               Exercise 3.3 
Solve the following systems of equations: 
1. 11x + 15y + 23 = 0 
7x – 2y – 20 = 0 
Sol: 
The given system of equation is 
? ?
? ?
11 15 23 0 ...
7 2 20 0 ...
x y i
x y ii
? ? ?
? ? ?
 
From (ii), we get 
2 7 20
7 20
2
yx
x
y
??
?
??
 
Substituting 
7 20
2
x
y
?
? in (i) we get 
7 20
11 15 23 0
2
105 300
11 23 0
2
22 105 300 46
0
2
127 254 0
127 254
254
2
127
x
x
x
x
xx
x
x
x
? ??
? ? ?
??
??
?
? ? ? ?
? ? ?
??
? ? ?
??
? ? ?
 
Page 2


 
 
 
 
 
               Exercise 3.3 
Solve the following systems of equations: 
1. 11x + 15y + 23 = 0 
7x – 2y – 20 = 0 
Sol: 
The given system of equation is 
? ?
? ?
11 15 23 0 ...
7 2 20 0 ...
x y i
x y ii
? ? ?
? ? ?
 
From (ii), we get 
2 7 20
7 20
2
yx
x
y
??
?
??
 
Substituting 
7 20
2
x
y
?
? in (i) we get 
7 20
11 15 23 0
2
105 300
11 23 0
2
22 105 300 46
0
2
127 254 0
127 254
254
2
127
x
x
x
x
xx
x
x
x
? ??
? ? ?
??
??
?
? ? ? ?
? ? ?
??
? ? ?
??
? ? ?
 
 
Putting 2 x ? in 
7 20
2
x
y
?
? we get 
7 2 20
2
14 20
2
6
2
3
y
??
??
?
?
?
?
??
 
Hence, the solution of the given system of equations is 2, 3. xy ? ? ? 
 
 
 
2. 3x – 7y + 10 = 0 
y – 2x – 3 = 0 
Sol: 
The given system of equation is 
? ?
? ?
3 7 10 0 ...
2 3 0 ...
x y i
y x ii
? ? ?
? ? ?
 
From (ii), we get 
23 yx ?? 
Substituting 23 yx ?? in (i) we get 
? ? 3 7 2 3 10 0
3 14 21 10 0
11 11
11
1
11
xx
xx
x
x
? ? ? ?
? ? ? ? ?
? ? ?
? ? ? ?
?
 
Putting 1 x ?? in 2 3, yx ?? we get 
? ? 2 1 3
23
1
1
y
y
? ? ? ? ?
? ? ?
?
??
 
Hence, the solution of the given system of equations is 1, 1. xy ? ? ? 
 
3. 0.4x + 0.3y = 1.7 
0.7x + 0.2y = 0.8 
Sol: 
Page 3


 
 
 
 
 
               Exercise 3.3 
Solve the following systems of equations: 
1. 11x + 15y + 23 = 0 
7x – 2y – 20 = 0 
Sol: 
The given system of equation is 
? ?
? ?
11 15 23 0 ...
7 2 20 0 ...
x y i
x y ii
? ? ?
? ? ?
 
From (ii), we get 
2 7 20
7 20
2
yx
x
y
??
?
??
 
Substituting 
7 20
2
x
y
?
? in (i) we get 
7 20
11 15 23 0
2
105 300
11 23 0
2
22 105 300 46
0
2
127 254 0
127 254
254
2
127
x
x
x
x
xx
x
x
x
? ??
? ? ?
??
??
?
? ? ? ?
? ? ?
??
? ? ?
??
? ? ?
 
 
Putting 2 x ? in 
7 20
2
x
y
?
? we get 
7 2 20
2
14 20
2
6
2
3
y
??
??
?
?
?
?
??
 
Hence, the solution of the given system of equations is 2, 3. xy ? ? ? 
 
 
 
2. 3x – 7y + 10 = 0 
y – 2x – 3 = 0 
Sol: 
The given system of equation is 
? ?
? ?
3 7 10 0 ...
2 3 0 ...
x y i
y x ii
? ? ?
? ? ?
 
From (ii), we get 
23 yx ?? 
Substituting 23 yx ?? in (i) we get 
? ? 3 7 2 3 10 0
3 14 21 10 0
11 11
11
1
11
xx
xx
x
x
? ? ? ?
? ? ? ? ?
? ? ?
? ? ? ?
?
 
Putting 1 x ?? in 2 3, yx ?? we get 
? ? 2 1 3
23
1
1
y
y
? ? ? ? ?
? ? ?
?
??
 
Hence, the solution of the given system of equations is 1, 1. xy ? ? ? 
 
3. 0.4x + 0.3y = 1.7 
0.7x + 0.2y = 0.8 
Sol: 
 
 
 
The given system of equation is 
? ?
? ?
0.4 0.3 1.7 ...
0.7 0.2 0.8 ...
x y i
x y ii
??
??
 
Multiplying both sides of (i) and (ii), by 10, we get 
? ?
? ?
4 3 17 ...
7 2 8 ...
x y iii
x y iv
??
??
 
From (iv), we get 
7 8 2
82
7
7
xy
y
x
??
?
?
 
Substituting 
82
7
y
x
?
? in (iii), we get 
82
4 3 17
7
32 8
3 17
7
32 29 17 7
y
y
y
y
y
? ??
??
??
??
?
? ? ?
? ? ? ?
 
29 119 32
29 87
87
3
29
y
y
y
? ? ?
??
? ? ?
 
Putting 3 y ? in 
82
,
7
y
x
?
? we get 
8 2 3
7
x
??
? 
86
7
14
7
2
?
?
?
?
 
Hence, the solution of the given system of equation is 2, 3. xy ?? 
 
4. 0.8
2
x
y ?? 
Sol: 
0.8
2
x
y ?? 
Page 4


 
 
 
 
 
               Exercise 3.3 
Solve the following systems of equations: 
1. 11x + 15y + 23 = 0 
7x – 2y – 20 = 0 
Sol: 
The given system of equation is 
? ?
? ?
11 15 23 0 ...
7 2 20 0 ...
x y i
x y ii
? ? ?
? ? ?
 
From (ii), we get 
2 7 20
7 20
2
yx
x
y
??
?
??
 
Substituting 
7 20
2
x
y
?
? in (i) we get 
7 20
11 15 23 0
2
105 300
11 23 0
2
22 105 300 46
0
2
127 254 0
127 254
254
2
127
x
x
x
x
xx
x
x
x
? ??
? ? ?
??
??
?
? ? ? ?
? ? ?
??
? ? ?
??
? ? ?
 
 
Putting 2 x ? in 
7 20
2
x
y
?
? we get 
7 2 20
2
14 20
2
6
2
3
y
??
??
?
?
?
?
??
 
Hence, the solution of the given system of equations is 2, 3. xy ? ? ? 
 
 
 
2. 3x – 7y + 10 = 0 
y – 2x – 3 = 0 
Sol: 
The given system of equation is 
? ?
? ?
3 7 10 0 ...
2 3 0 ...
x y i
y x ii
? ? ?
? ? ?
 
From (ii), we get 
23 yx ?? 
Substituting 23 yx ?? in (i) we get 
? ? 3 7 2 3 10 0
3 14 21 10 0
11 11
11
1
11
xx
xx
x
x
? ? ? ?
? ? ? ? ?
? ? ?
? ? ? ?
?
 
Putting 1 x ?? in 2 3, yx ?? we get 
? ? 2 1 3
23
1
1
y
y
? ? ? ? ?
? ? ?
?
??
 
Hence, the solution of the given system of equations is 1, 1. xy ? ? ? 
 
3. 0.4x + 0.3y = 1.7 
0.7x + 0.2y = 0.8 
Sol: 
 
 
 
The given system of equation is 
? ?
? ?
0.4 0.3 1.7 ...
0.7 0.2 0.8 ...
x y i
x y ii
??
??
 
Multiplying both sides of (i) and (ii), by 10, we get 
? ?
? ?
4 3 17 ...
7 2 8 ...
x y iii
x y iv
??
??
 
From (iv), we get 
7 8 2
82
7
7
xy
y
x
??
?
?
 
Substituting 
82
7
y
x
?
? in (iii), we get 
82
4 3 17
7
32 8
3 17
7
32 29 17 7
y
y
y
y
y
? ??
??
??
??
?
? ? ?
? ? ? ?
 
29 119 32
29 87
87
3
29
y
y
y
? ? ?
??
? ? ?
 
Putting 3 y ? in 
82
,
7
y
x
?
? we get 
8 2 3
7
x
??
? 
86
7
14
7
2
?
?
?
?
 
Hence, the solution of the given system of equation is 2, 3. xy ?? 
 
4. 0.8
2
x
y ?? 
Sol: 
0.8
2
x
y ?? 
 
And 
7
10
2
y
x
?
?
 
72
2 1.6 10
2
x y and
xy
?
? ? ? ?
?
 
2 1.6 7 10 5 x y and x y ? ? ? ? 
Multiply first equation by 10 
10 20 16 10 5 7 x y and x y ? ? ? ? 
Subtracting the two equations 
15 9
93
15 5
3 6 2
1.6 2 1.6
5 5 5
y
y
x
?
??
??
? ? ? ? ?
??
??
 
Solution is 
23
,
55
??
??
??
 
 
 
 
5. 7(y + 3) – 2 (x + 3) = 14 
4(y – 2) + 3 (x – 3) = 2 
Sol: 
The given system of equations id 
? ? ? ? ? ?
? ? ? ? ? ?
7 3 2 3 14 ...
4 2 3 3 2 ...
y x i
y x ii
? ? ? ?
? ? ? ?
 
From (i), we get 
7 21 2 4 14
7 14 4 21 2
23
7
xx
yx
x
y
? ? ? ?
? ? ? ? ?
?
??
 
From (ii), we get 
4 8 3 9 2 yx ? ? ? ? 
? ?
4 3 17 2 0
4 3 19 0 ...
yx
y x iii
? ? ? ? ?
? ? ? ?
 
Substituting 
23
7
x
y
?
? in (iii), we get 
Page 5


 
 
 
 
 
               Exercise 3.3 
Solve the following systems of equations: 
1. 11x + 15y + 23 = 0 
7x – 2y – 20 = 0 
Sol: 
The given system of equation is 
? ?
? ?
11 15 23 0 ...
7 2 20 0 ...
x y i
x y ii
? ? ?
? ? ?
 
From (ii), we get 
2 7 20
7 20
2
yx
x
y
??
?
??
 
Substituting 
7 20
2
x
y
?
? in (i) we get 
7 20
11 15 23 0
2
105 300
11 23 0
2
22 105 300 46
0
2
127 254 0
127 254
254
2
127
x
x
x
x
xx
x
x
x
? ??
? ? ?
??
??
?
? ? ? ?
? ? ?
??
? ? ?
??
? ? ?
 
 
Putting 2 x ? in 
7 20
2
x
y
?
? we get 
7 2 20
2
14 20
2
6
2
3
y
??
??
?
?
?
?
??
 
Hence, the solution of the given system of equations is 2, 3. xy ? ? ? 
 
 
 
2. 3x – 7y + 10 = 0 
y – 2x – 3 = 0 
Sol: 
The given system of equation is 
? ?
? ?
3 7 10 0 ...
2 3 0 ...
x y i
y x ii
? ? ?
? ? ?
 
From (ii), we get 
23 yx ?? 
Substituting 23 yx ?? in (i) we get 
? ? 3 7 2 3 10 0
3 14 21 10 0
11 11
11
1
11
xx
xx
x
x
? ? ? ?
? ? ? ? ?
? ? ?
? ? ? ?
?
 
Putting 1 x ?? in 2 3, yx ?? we get 
? ? 2 1 3
23
1
1
y
y
? ? ? ? ?
? ? ?
?
??
 
Hence, the solution of the given system of equations is 1, 1. xy ? ? ? 
 
3. 0.4x + 0.3y = 1.7 
0.7x + 0.2y = 0.8 
Sol: 
 
 
 
The given system of equation is 
? ?
? ?
0.4 0.3 1.7 ...
0.7 0.2 0.8 ...
x y i
x y ii
??
??
 
Multiplying both sides of (i) and (ii), by 10, we get 
? ?
? ?
4 3 17 ...
7 2 8 ...
x y iii
x y iv
??
??
 
From (iv), we get 
7 8 2
82
7
7
xy
y
x
??
?
?
 
Substituting 
82
7
y
x
?
? in (iii), we get 
82
4 3 17
7
32 8
3 17
7
32 29 17 7
y
y
y
y
y
? ??
??
??
??
?
? ? ?
? ? ? ?
 
29 119 32
29 87
87
3
29
y
y
y
? ? ?
??
? ? ?
 
Putting 3 y ? in 
82
,
7
y
x
?
? we get 
8 2 3
7
x
??
? 
86
7
14
7
2
?
?
?
?
 
Hence, the solution of the given system of equation is 2, 3. xy ?? 
 
4. 0.8
2
x
y ?? 
Sol: 
0.8
2
x
y ?? 
 
And 
7
10
2
y
x
?
?
 
72
2 1.6 10
2
x y and
xy
?
? ? ? ?
?
 
2 1.6 7 10 5 x y and x y ? ? ? ? 
Multiply first equation by 10 
10 20 16 10 5 7 x y and x y ? ? ? ? 
Subtracting the two equations 
15 9
93
15 5
3 6 2
1.6 2 1.6
5 5 5
y
y
x
?
??
??
? ? ? ? ?
??
??
 
Solution is 
23
,
55
??
??
??
 
 
 
 
5. 7(y + 3) – 2 (x + 3) = 14 
4(y – 2) + 3 (x – 3) = 2 
Sol: 
The given system of equations id 
? ? ? ? ? ?
? ? ? ? ? ?
7 3 2 3 14 ...
4 2 3 3 2 ...
y x i
y x ii
? ? ? ?
? ? ? ?
 
From (i), we get 
7 21 2 4 14
7 14 4 21 2
23
7
xx
yx
x
y
? ? ? ?
? ? ? ? ?
?
??
 
From (ii), we get 
4 8 3 9 2 yx ? ? ? ? 
? ?
4 3 17 2 0
4 3 19 0 ...
yx
y x iii
? ? ? ? ?
? ? ? ?
 
Substituting 
23
7
x
y
?
? in (iii), we get 
 
23
4 3 19 0
7
8 12
3 19 0
7
8 12 21 133 0
29 145 0
29 145
145
5
29
x
x
x
x
xx
x
x
x
? ??
? ? ?
??
??
?
? ? ? ?
? ? ? ? ?
? ? ?
??
? ? ?
 
Putting 5 x ? in 
23
,
7
x
y
?
? we get 
2 5 7
7
y
??
? 
10 3
7
7
7
1
1 y
?
?
?
?
??
 
Hence, the solution of the given system of equations is 5, 1. xy ?? 
 
6. 
?? 7
+
?? 3
= 5 
?? 2
-
?? 9
= 6  
Sol: 
The given system of equation is 
? ?
? ?
5 ...
73
6 ...
29
xy
i
xy
ii
??
??
 
From (i), we get 
37
5
21
3 7 105
3 105 7
105 7
3
xy
xy
xy
y
x
?
?
? ? ?
? ? ?
?
??
 
From (ii), we get 
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