5. Chapter 5. Linear Equations NIOS Class 10 Mathematics Textbook Solutions
Chapter 5. Linear Equations NIOS Class 10 Mathematics Textbook Solutions for Exam Preparation
Chapter 5. LINEAR EQUATIONS
CHECK YOUR PROGRESS 5.1
1. Which of the following are linear equations in one variable?
(i) 
(ii) 
(iii) 
(iv) 
Solution: (i)
[It is a linear equation in x as the exponent of x is 1.]
2. Which of the following are linear equations in two variables:
(i) 
(ii) 
(iii) 
Solution: (i)
[ It is a linear equation in two variables y and x.]
CHECK YOUR PROGRESS 5.2
Form a linear equation using suitable variables for the following situations:
1. Twice a number subtracted from 15 is 7.
Solution: Let x be the number .
A/Q, 
2. A motor boat uses 0.1 litres of fuel for every kilometer. One day, it made a trip of x km. Form an equation in x, if the total consumption of fuel was 10 litres.
Solution: Given, x be the distance traveled (in kilometers) .
A/Q, 
3. The length of rectangle is twice its width. The perimeter of rectangle is 96m. [Assume width of rectangle as y m]
Solution: Given, y be the width of the rectangle and the length will be 2y (in meter) .
A/Q, 
4. After 15 years, Salma will be four times as old as she is now. [Assume present age of Salma as t years]
Solution: Given, t be the present age of Salma .
A/Q, 
CHECK YOUR PROGRESS 5.3
Solve the following equations:
1. 
2. 
3. 
4. 
5. 
Solution: 1. We have,
2. We have,
3. We have,
4. We have, 
5. We have, 
CHECK YOUR PROGRESS 5.4
1. The sum of two numbers is 85. If one number exceeds the other by 7, find the numbers.
Solution: Let, x be the number and the other number will be (x+7) .
A/Q, 
Therefore, the number are 39 and 46 (= 39 + 7) .
2. The age of father is 20 years more than twice the age of the son. If sum of their ages is 65 years, find the age of the son and the father.
Solution: Let the son's age be x years.
Father's age 
years.
A/Q, 
So, the son’s age is 15 years.
Father’s age = 2 × 15 + 20= 50 years.
3. The length of a rectangle is twice its breadth. If perimeter of rectangle is 66 cm, find its length and breadth.
Solution: Let the breadth of the rectangle be y cm and the length = 2y cm.
We know that , Perimeter=2(Length + Breadth)
A/Q, 
Therefore, the breadth is 11 cm and the length = 2×11=22 cm .
4. In a class, the number of boys is 2/5 of the number of girls. Find the number of girls in the class, if the number of boys is 10.
Solution: Let the number of girls be x
A/Q, 
So, the number of girls is 25.
CHECK YOUR PROGRESS 5.5
1. Form linear equations in two variables using suitable variables for the unknowns.
(i) The perimeter of a rectangle is 98 cm. [Take length as x and breadth as y.]
(ii) The age of father is 10 years more than twice the age of son.
(iii) A number is 10 more than the other number.
(iv) The cost of 2kg apples and 3 kg oranges is Rs. 120. [Take x and y as the cost per kg of apples and oranges respectively.]
Solution: (i) Let, x be the length and y be breadth .
A/Q, 
(ii) Let the son’s age be x years and the father’s age be y years.
A/Q, 
(iii) Let the first number be x and the second number be y .
A/Q, 
(iv) Given, the cost per kg of apples be x rupees and the cost per kg of oranges be y rupees.
A/Q, 
Write True or False for the following:
2. 
is a solution of the equation 
.
Solution: The equation is 
Here, 
LHS: 
RHS
3. 
is a solution of the equation 
Solution: The equation is 
Here, 
LHS : 
CHECK YOUR PROGRESS 5.6
1. Plot the following points in the cartesian plane:
(i) (3, 4) (ii) (–3, –2) (iii) (–2, 1) (iv) (2, –3) (v) (4, 0) (vi) (0, –3)
Solution:
2. Draw the graph of each of the following linear equations in two variables:
(i) 
(ii) 
(iii) 
(iv) 
Solution: (i) We have, 
|
x |
5 |
0 |
2 |
|
y |
0 |
5 |
3 |
(ii) We have, 
|
x |
0 |
2 |
– 2 |
|
y |
3 |
0 |
6 |
(iii) We have, 
|
x |
0 |
3 |
2 |
|
y |
6 |
0 |
2 |
(iv) We have,
|
x |
– 1 |
2 |
– 4 |
|
y |
3 |
– 2 |
8 |
CHECK YOUR PROGRESS 5.7
Solve the following system of equations graphically. Also, tell whether these have unique solution, infinitely many solutions or no solution.
1. 
2. 
3. 
4. 
5. 
Solution: 1.
We have, 
|
x |
3 |
0 |
4 |
|
y |
0 |
– 3 |
1 |
and 
|
x |
0 |
5 |
3 |
|
y |
5 |
0 |
2 |
2.
We have, 
|
x |
– 1 |
2 |
– 4 |
|
y |
1 |
– 1 |
3 |
and
|
x |
1 |
2 |
3 |
|
y |
– 4 |
– 1 |
2 |
3.
We have, 
|
x |
6 |
0 |
2 |
|
y |
0 |
3 |
2 |
and 
|
x |
6 |
0 |
2 |
|
y |
0 |
3 |
2 |
4.
We have.
|
x |
0 |
2 |
4 |
|
y |
3 |
0 |
– 3 |
And 
|
x |
1 |
– 1 |
3 |
|
y |
3 |
6 |
0 |
5.
We have, 
|
x |
0 |
1 |
2 |
|
y |
5 |
3 |
1 |
and 
|
x |
0 |
2 |
4 |
|
y |
4 |
1 |
– 2 |
CHECK YOUR PROGRESS 5.8
Solve the following system of equations by substitution method:
1. 
2. 
3. 
4. 
Solution: 1. 
We have, 
And 
Putting 
in (i) , we get 
2. 
We have, 
And 
Putting 
in (i) , we get 
3.
We have, 
And 
Putting 
in (i) , we get
4. 
We have, 
And 
Putting 
in (i) , We get
CHECK YOUR PROGRESS 5.9
Solve the following systems of equations by elimination method:
1. 
2. 
3. 
4. 
5. 
6. 
Solution: 1. We have, 
And 
Putting 
in (i) , we get 
2. We have, 
And 
Putting 
in (i) , we get 
3. We have, 
And 
Putting 
in (i) , we get 
4. We have, 
Putting 
in (i) , we get 
5. We have, 
And 
Putting
in (i) , we get 
6. We have, 
And 
Putting 
in (i) , we get 
CHECK YOUR PROGRESS 5.10
1. Rahim's father is three times as old as Rahim. If sum of their ages is 56 years, find their ages.
Solution: Let, Rahim’s age be x years and his father’s age be y years.
A/Q, 
And 
[ from (i) ]
Putting 
in equation (i) , we get 
Therefore, Rahim is 14 years old and his father is 42 years old.
2. Rita has 18m of cloth. She cut it into two pieces in such a way that one piece is 4 m longer than the other. What is the length of shorter piece.
Solution: Let, x and y be the length of the shorter piece and the length of the longer piece (in meters).
A/Q, 
(i)
And 
[ from (i) ]
Putting 
in (i) , we get 
Therefore, the shorter piece is 7 m​ and the longer piece is 11 m .
3. A total of Rs 50000 is to be distributed among 200 persons as prizes. A prize is either Rs 500 or Rs 100. Find the number of each type of prizes.
Solution: Let x be the number of Rs 500 prizes and y be the number of Rs 100 prizes.
A/Q, 
(i)
And 
Putting 
in (i) , we get 
Therefore, the number of Rs 500 prizes is 75 and the number of Rs 100 prizes is 125 .
4. A purse contain Rs 2500 in notes of denominations of 100 and 50. If the number of 100 rupee notes is one more than that of 50 rupee notes, find the number of notes of each denomination.
Solution: Let, x and y be the number of 100 rupee notes and 50 rupee notes respectively .
A/Q, 
(i)
And 
Putting 
in (i) , we get 
Therefore, the number of 100 rupee notes and 50 rupee notes are 17 and 16 respectively .
TERMINAL EXERCISE
1. Choose the correct option:
(i) Which one of the following is a linear equation in one variable?
(A) 
(B) 
(C) 
(D) 
Answer: (B) 
[ It is a linear equation in t as the exponent of t is 1 .]
(ii) Which one of the following is not a linear equation?
(A) 
(B) 
(C) 
(D) 
Answer: (C) 
(iii) Which of the following numbers is the solution of the equation 
?
(A) 6 (B) 12 (C) 13 (D) 21
Answer: (A) 6
[ We have, 
]
(iv) The value of x, for which the equation 
is satisfied, is:
(A) 4.5 (B) 3 (C) 2.25 (D) 0.5
Answer: (C) 2.25
[ We have, 
]
(v) The equation 
has
(A) no solution (B) unique solution (C) two solutions (D) infinitely many solutions
Answer: (D) infinitely many solutions .
[ A linear equation in two variables will have infinitely many solutions. ]
2. Solve each of the following equations
(i) 
(ii) 
(iii) 
(iv) 
Solution: (i) We have, 
(ii) We have, 
(iii) We have,
(iv) We have, 
3. A certain number increased by 8 equals 26. Find the number.
Solution: Let the unknown number be x .
A/Q, 
Therefore, the number is 18 .
4. Present ages of Reena and Meena are in the ration 4 : 5. After 8 years, the ratio of their ages will be 5 : 6. Find their present ages.
Solution: Let 4x and 5y be the present ages of Reena and Meena , respectively .
After 8 years, Reena's age = 4x+8 and Meena's age = 5x+8
A/Q, 
Therefore, Reena's present age 
years
And Meena's present age 
years .
5. The denominator of a rational number is greater than its numerator by 8. If the denominaor is decreased by 1 and numerator is increased by 17, the number obtained is 3/2 . Find the rational number
Solution: Let , x and y be the numerator and the denominator of the rational number, respectively.
The rational number is 
.
A/Q, 
(i)
and 
Putting 
in (i) , we get 
Therefore, the rational number is 
.
6. Solve the following system of equations graphically:
(i) 
(ii) 
(iii) 
(iv) 
Solution: (i)
We have, 
|
x |
7 |
– 1 |
– 2 |
|
y |
0 |
5 |
3 |
And 
|
x |
– 2 |
0 |
– 1 |
|
y |
0 |
– 2 |
1 |
(ii)
We have, 
|
x |
6 |
0 |
3 |
|
y |
0 |
8 |
4 |
and 
|
x |
0 |
– 3 |
3 |
|
y |
2 |
0 |
4 |
(iii)
We have, 
|
x |
6 |
0 |
3 |
|
y |
0 |
2 |
1 |
and 
|
x |
2 |
1 |
3 |
|
y |
– 1 |
– 3 |
1 |
(iv)
We have, 
|
x |
0 |
1 |
2 |
|
y |
– 1 |
1 |
3 |
and 
|
x |
4 |
3 |
2 |
|
y |
4 |
5 |
6 |
7. Solve the following system of equations :
(i) 
(ii) 
(iii) 
(iv) 
Solution: (i)
We have, 
(i)
and 
Putting 
in (i) , we get 
(ii) 
We have, 
and 
Putting in (i) , we get
(iii)
We have, 
and 
Putting 
in (i) , we get 
(iv)
We have, 
And 
Putting 
in (i) , we get 
8. The sum of the digits of a two-digit number is 11. If the digits are reversed, the new number is 27 less than the original number. Find the original number.
Solution: Let x is the tens digit and y is the units digit.
The original two-digit number be 
.
The digits are reversed, then the new number is 
.
A/Q, 
(i)
And 
[ From (i)]
Putting 
in equation (i) , we get 
The original number 
.
9. Three years ago Atul's age was four times Parul's age. After 5 years from now, Atul's age will be two times Parul's age. Find their present ages.
Solution: Let Atul’s present age be x and Parul’s present age be y .
Three years ago, Atul’s age was 
and Parul’s age was 
.
After 5 years from now , Atul’s age will be 
and Parul’s age will be 
.
A/Q, 
(i)
And 
[ from (i)]
Putting 
in equation (i) , we get 
Atul’s present age is 19 years, and Parul’s present age is 7 years.
10. The perimeter of a rectangular plot of land is 32 m. If the length is increased by 2m and breadth is decreased by 1 m, the area of the plot remains the same. Find the length and breadth of the plot.
Solution: Let, x and y be the length and breadth of the plot respectively (in m).
So, the area of the plot is xy .
A/Q, 
(i)
And 
Putting 
in equation (i) , we get 
Therefore, the length and the breadth are 10 m and 6 m respectively .
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