RS Aggarwal Solutions: Exercise 3F - Linear Equations in two variables | Extra Documents, Videos & Tests for Class 10 PDF Download

 Page 1


Exercise  3F 
1. Write the number of solutions of the following pair of linear equations:
x + 2y -8=0,
2x + 4y = 16
Sol:
The given equations are
x + 2y 
2x + 4y 
Which is of the form a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0, where
a1 = 1, b1 = 2, c1 = -8, a2 = 2, b2 = 4 and c2 = -18
Page 2


Exercise  3F 
1. Write the number of solutions of the following pair of linear equations:
x + 2y -8=0,
2x + 4y = 16
Sol:
The given equations are
x + 2y 
2x + 4y 
Which is of the form a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0, where
a1 = 1, b1 = 2, c1 = -8, a2 = 2, b2 = 4 and c2 = -18
 
Now
=
= =
= =
= = =
Thus, the pair of linear equations are coincident and therefore has infinitely many solutions.
2. Find the value of k for which the system of linear equations has an infinite number of
solutions.
2x + 3y 7 = 0,
(k 1)x + (k + 2)y=3k
Sol:
The given equations are
2x + 3y 
(k 1)x + (k + 2)y 
Which is of the form a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0, where
a1 = 2, b1 = 3, c1 = -7, a2 = k 1, b2 = k + 2 and c2 = -3k
For the given pair of linear equations to have infinitely many solutions, we must have
= =
= =
= , = and =
2(k + 2) = 3(k 1), 9k = 7k + 14 and 6k = 7k 7
k = 7, k = 7 and k = 7
Hence, k = 7.
3. Find the value of k for which the system of linear equations has an infinite number of
solutions.
10x + 5y (k 5) = 0,
20x + 10y k = 0.
Sol:
The given pair of linear equations are
10x + 5y (k
20x + 10y 
Which is of the form a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0, where
a1 = 10, b1 = 5, c1 = -(k 5), a2 = 20, b2 = 10 and c2 = -k
For the given pair of linear equations to have infinitely many solutions, we must have
Page 3


Exercise  3F 
1. Write the number of solutions of the following pair of linear equations:
x + 2y -8=0,
2x + 4y = 16
Sol:
The given equations are
x + 2y 
2x + 4y 
Which is of the form a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0, where
a1 = 1, b1 = 2, c1 = -8, a2 = 2, b2 = 4 and c2 = -18
 
Now
=
= =
= =
= = =
Thus, the pair of linear equations are coincident and therefore has infinitely many solutions.
2. Find the value of k for which the system of linear equations has an infinite number of
solutions.
2x + 3y 7 = 0,
(k 1)x + (k + 2)y=3k
Sol:
The given equations are
2x + 3y 
(k 1)x + (k + 2)y 
Which is of the form a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0, where
a1 = 2, b1 = 3, c1 = -7, a2 = k 1, b2 = k + 2 and c2 = -3k
For the given pair of linear equations to have infinitely many solutions, we must have
= =
= =
= , = and =
2(k + 2) = 3(k 1), 9k = 7k + 14 and 6k = 7k 7
k = 7, k = 7 and k = 7
Hence, k = 7.
3. Find the value of k for which the system of linear equations has an infinite number of
solutions.
10x + 5y (k 5) = 0,
20x + 10y k = 0.
Sol:
The given pair of linear equations are
10x + 5y (k
20x + 10y 
Which is of the form a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0, where
a1 = 10, b1 = 5, c1 = -(k 5), a2 = 20, b2 = 10 and c2 = -k
For the given pair of linear equations to have infinitely many solutions, we must have
 
= =
= =
=
2k 10 = k k = 10
Hence, k = 10.
4. Find the value of k for which the system of linear equations has an infinite number of
solutions.
2x + 3y=9,
6x + (k 2)y =(3k 2
Sol:
The given pair of linear equations are
2x + 3y 9 = 0                          
6x + (k 2)y (3k
Which is of the form a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0, where
a1 = 2, b1 = 3, c1 = -9, a2 = 6, b2 = k 2 and c2 = -(3k 2)
For the given pair of linear equations to have infinitely many solutions, we must have
=
=
= ,
k = 11,
k = 11, 3(3k 2)
Hence, k = 11.
5. Write the number of solutions of the following pair of linear equations:
x + 3y 4 = 0, 2x + 6y 7 = 0.
Sol:
The given pair of linear equations are
x + 3y 
2x + 6y 
Which is of the form a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0, where
a1 = 1, b1 = 3, c1 = -4, a2 = 2, b2 = 6 and c2 = -7
Now
=
= =
Page 4


Exercise  3F 
1. Write the number of solutions of the following pair of linear equations:
x + 2y -8=0,
2x + 4y = 16
Sol:
The given equations are
x + 2y 
2x + 4y 
Which is of the form a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0, where
a1 = 1, b1 = 2, c1 = -8, a2 = 2, b2 = 4 and c2 = -18
 
Now
=
= =
= =
= = =
Thus, the pair of linear equations are coincident and therefore has infinitely many solutions.
2. Find the value of k for which the system of linear equations has an infinite number of
solutions.
2x + 3y 7 = 0,
(k 1)x + (k + 2)y=3k
Sol:
The given equations are
2x + 3y 
(k 1)x + (k + 2)y 
Which is of the form a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0, where
a1 = 2, b1 = 3, c1 = -7, a2 = k 1, b2 = k + 2 and c2 = -3k
For the given pair of linear equations to have infinitely many solutions, we must have
= =
= =
= , = and =
2(k + 2) = 3(k 1), 9k = 7k + 14 and 6k = 7k 7
k = 7, k = 7 and k = 7
Hence, k = 7.
3. Find the value of k for which the system of linear equations has an infinite number of
solutions.
10x + 5y (k 5) = 0,
20x + 10y k = 0.
Sol:
The given pair of linear equations are
10x + 5y (k
20x + 10y 
Which is of the form a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0, where
a1 = 10, b1 = 5, c1 = -(k 5), a2 = 20, b2 = 10 and c2 = -k
For the given pair of linear equations to have infinitely many solutions, we must have
 
= =
= =
=
2k 10 = k k = 10
Hence, k = 10.
4. Find the value of k for which the system of linear equations has an infinite number of
solutions.
2x + 3y=9,
6x + (k 2)y =(3k 2
Sol:
The given pair of linear equations are
2x + 3y 9 = 0                          
6x + (k 2)y (3k
Which is of the form a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0, where
a1 = 2, b1 = 3, c1 = -9, a2 = 6, b2 = k 2 and c2 = -(3k 2)
For the given pair of linear equations to have infinitely many solutions, we must have
=
=
= ,
k = 11,
k = 11, 3(3k 2)
Hence, k = 11.
5. Write the number of solutions of the following pair of linear equations:
x + 3y 4 = 0, 2x + 6y 7 = 0.
Sol:
The given pair of linear equations are
x + 3y 
2x + 6y 
Which is of the form a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0, where
a1 = 1, b1 = 3, c1 = -4, a2 = 2, b2 = 6 and c2 = -7
Now
=
= =
 
= =
=
Thus, the pair of the given linear equations has no solution.
6. Find the values of k for which the system of equations 3x + ky = 0,
2x y = 0 has a unique solution.
Sol:
The given pair of linear equations are
2x 
Which is of the form a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0, where
a1 = 3, b1 = k, c1 = 0, a2 = 2, b2 = -1 and c2 = 0
For the system to have a unique solution, we must have
=
Hence, .
7. The difference of two numbers is 5 and the difference between their squares is 65. Find the
numbers.
Sol:
Let the numbers be x and y, where x y.
Then as per the question
x
x
2
y
2
Dividing (ii) by (i), we get
=
= 13
Now, adding (i) and (ii), we have
2x = 18 x = 9
Substituting x = 9 in (iii), we have
9 + y = 13 y = 4
Hence, the numbers are 9 and 4.
Page 5


Exercise  3F 
1. Write the number of solutions of the following pair of linear equations:
x + 2y -8=0,
2x + 4y = 16
Sol:
The given equations are
x + 2y 
2x + 4y 
Which is of the form a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0, where
a1 = 1, b1 = 2, c1 = -8, a2 = 2, b2 = 4 and c2 = -18
 
Now
=
= =
= =
= = =
Thus, the pair of linear equations are coincident and therefore has infinitely many solutions.
2. Find the value of k for which the system of linear equations has an infinite number of
solutions.
2x + 3y 7 = 0,
(k 1)x + (k + 2)y=3k
Sol:
The given equations are
2x + 3y 
(k 1)x + (k + 2)y 
Which is of the form a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0, where
a1 = 2, b1 = 3, c1 = -7, a2 = k 1, b2 = k + 2 and c2 = -3k
For the given pair of linear equations to have infinitely many solutions, we must have
= =
= =
= , = and =
2(k + 2) = 3(k 1), 9k = 7k + 14 and 6k = 7k 7
k = 7, k = 7 and k = 7
Hence, k = 7.
3. Find the value of k for which the system of linear equations has an infinite number of
solutions.
10x + 5y (k 5) = 0,
20x + 10y k = 0.
Sol:
The given pair of linear equations are
10x + 5y (k
20x + 10y 
Which is of the form a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0, where
a1 = 10, b1 = 5, c1 = -(k 5), a2 = 20, b2 = 10 and c2 = -k
For the given pair of linear equations to have infinitely many solutions, we must have
 
= =
= =
=
2k 10 = k k = 10
Hence, k = 10.
4. Find the value of k for which the system of linear equations has an infinite number of
solutions.
2x + 3y=9,
6x + (k 2)y =(3k 2
Sol:
The given pair of linear equations are
2x + 3y 9 = 0                          
6x + (k 2)y (3k
Which is of the form a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0, where
a1 = 2, b1 = 3, c1 = -9, a2 = 6, b2 = k 2 and c2 = -(3k 2)
For the given pair of linear equations to have infinitely many solutions, we must have
=
=
= ,
k = 11,
k = 11, 3(3k 2)
Hence, k = 11.
5. Write the number of solutions of the following pair of linear equations:
x + 3y 4 = 0, 2x + 6y 7 = 0.
Sol:
The given pair of linear equations are
x + 3y 
2x + 6y 
Which is of the form a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0, where
a1 = 1, b1 = 3, c1 = -4, a2 = 2, b2 = 6 and c2 = -7
Now
=
= =
 
= =
=
Thus, the pair of the given linear equations has no solution.
6. Find the values of k for which the system of equations 3x + ky = 0,
2x y = 0 has a unique solution.
Sol:
The given pair of linear equations are
2x 
Which is of the form a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0, where
a1 = 3, b1 = k, c1 = 0, a2 = 2, b2 = -1 and c2 = 0
For the system to have a unique solution, we must have
=
Hence, .
7. The difference of two numbers is 5 and the difference between their squares is 65. Find the
numbers.
Sol:
Let the numbers be x and y, where x y.
Then as per the question
x
x
2
y
2
Dividing (ii) by (i), we get
=
= 13
Now, adding (i) and (ii), we have
2x = 18 x = 9
Substituting x = 9 in (iii), we have
9 + y = 13 y = 4
Hence, the numbers are 9 and 4.
 
8. The cost of 5 pens and 8 pencils together cost Rs. 120 while 8 pens and 5 pencils together
cost Rs. 153. Find the cost of a 1 pen and that of a 1pencil.
Sol:
Then as per the question
5x + 8y = 120               
Adding (i) and (ii), we get
13x + 13y = 273
Subtracting (i) from (ii), we get
3x 3y = 33
x
Now, adding (iii) and (iv), we get
2x = 32 x = 16
Substituting x = 16 in (iii), we have
16 + y = 21 y = 5
9. The sum of two numbers is 80. The larger number exceeds four times the smaller one by 5.
Find the numbers.
Sol:
Let the larger number be x and the smaller number be y.
Then as per the question
x = 4y + 5
x
Subtracting (ii) from (i), we get
5y = 75 y = 15
Now, putting y = 15 in (i), we have
x + 15 = 80 x = 65
Hence, the numbers are 65 and 15.
10. A number consists of two digits whose sum is 10. If 18 is subtracted form the number, its
digits are reversed. Find the number.
Sol:
Let the ones digit and tens digit be x and y respectively.
Then as per the question