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Pair of Linear Equations in Two Variables - 3 RD Sharma Solutions | Mathematics (Maths) Class 10 PDF Download

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 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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FAQs on Pair of Linear Equations in Two Variables - 3 RD Sharma Solutions - Mathematics (Maths) Class 10

1. What is the definition of a pair of linear equations in two variables?
Ans. A pair of linear equations in two variables is a set of two equations that involve two variables and are of degree one. The general form of such equations is ax + by = c, where a, b, and c are constants.
2. How can I solve a pair of linear equations in two variables?
Ans. There are several methods to solve a pair of linear equations in two variables, such as the substitution method, elimination method, and graphical method. The choice of method depends on the given equations and personal preference.
3. Is it always possible to find a solution for a pair of linear equations in two variables?
Ans. Not always. If the given pair of equations represents parallel lines, there will be no solution as parallel lines do not intersect. Similarly, if the equations represent the same line, there will be infinitely many solutions.
4. What is the importance of solving a pair of linear equations in two variables?
Ans. Solving a pair of linear equations in two variables helps in finding the common solution(s) of the equations. These solutions can be used to determine the values of the variables in real-life situations, such as finding the intersection point of two lines or solving problems related to cost and revenue.
5. Can a pair of linear equations in two variables have more than one solution?
Ans. Yes, a pair of linear equations in two variables can have infinitely many solutions. This happens when the equations represent the same line. In such cases, any point on the line will satisfy both equations, resulting in infinitely many solutions.
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