# Form the pair of linear equations in the following problems and find their solutions (if they exist) by any algebraic method:

(i) A part of monthly hostel charges is fixed and the remaining depends on the number of days one has taken food in the mess. When student A takes food for 20 days, she has to pay ₹ 1000 as hostel charges whereas a student B, who takes food for 26 days, pays ₹ 1180 as hostel charges. Find the fixed charges and the cost of food per day.

(ii) A fraction becomes 1/3 when 1 is subtracted from the numerator and it becomes 1/4 when 8 is added to its denominator. Find the fraction.

(iii) Yash scored 40 marks in a test, getting 3 marks for each right answer and losing 1 mark for each wrong answer. Had 4 marks been awarded for each correct answer and 2 marks been deducted for each incorrect answer, then Yash would have scored 50 marks. How many questions were there in the test?

(iv) Places A and B are 100 km apart on a highway. One car starts from A and another from B at the same time. If the cars travel in the same direction at different speeds, they meet in 5 hours. If they travel towards each other, they meet in 1 hour. What are the speeds of the two cars?

(v) The area of a rectangle gets reduced by 9 square units, if its length is reduced by 5 units and breadth is increased by 3 units. If we increase the length by 3 units and the breadth by 2 units, the area increases by 67 square. Find the dimensions of the rectangle

**Solution:**

Let us assume one variable equal to x and another be y. Then two linear equations can be formed based on given conditions, which can be easily solved.

(i) Let x be the fixed charge of the food and y be the charge for food per day. According to the given information,

When student A, takes food for 20 days, pays ₹ 1000 as hostel charges.

x + 20 y = 1000 ....(1)

When student B, who takes food for 26 days, pays ₹ 1180 as hostel charges.

x + 26 y = 1180 ....(2)

Subtracting equation (1) from equation (2), we obtain

6 y = 180

y = 180/6

y = 30

Substituting y = 30 in equation (1), we obtain

x + 20 × 30 = 1000

x = 1000 - 600

x = 400

Equations are x + 20 y = 1000 and x + 26 y = 1180 where x is the fixed charge of the food and y is the charge for food per day.

Hence, the fixed charge is ₹ 400 and charge per day is ₹ 30

(ii) Let the numerator be x and denominator be y, thus the fraction be x/y

According to the given information,

When 1 is subtracted from the numerator

(x -1)y = 1/3

3x - 3 = y

3x - y = 3 ....(1)

When 8 is added to the denominator,

x/(y + 8) = 1/4

4x = y + 8

4x - y = 8 ....(2)

Subtracting equation (1) from equation (2), we obtain

x = 5

Putting x = 5 in equation (1), we obtain

3 × 5 - y = 3

y = 15 - 3

y = 12

Equations are 3x - y = 3 and 4x - y = 8 where the numerator of the fraction is x, and the denominator is y.

Hence, the fraction is 5/12

(iii) Let the number of right answers and wrong answers be x and y respectively. Therefore, the total number of questions be ( x + y )

According to the given information,

3x - y = 40 ....(1)

4x - 2 y = 50

2x - y = 25 ....(2)

Subtracting equation (2) from equation (1), we obtain

x = 15 ....(3)

Substituting this in equation (2), we obtain

2 × 15 - y = 25

y = 30 - 25

y = 5

Equations are 3x - y = 40 and 2x - y = 25 where the number of right and wrong answers are x and y respectively.

number of right answers = 15 and number of wrong answers = 5

Hence, the total number of questions = 20

(iv) Let the speed of the 1^{st} car and 2^{nd} car be u km/h and v km/h respectively.

According to the given information,

When the cars travel in the same direction at different speeds, they meet in 5 hours.

therefore, distance travelled by 1^{st} car = 5u km

and distance travelled by 2^{nd} car = 5v km

5u - 5v = 100

5(u - v) = 100

u - v = 20 ....(1)

When the cars travel towards each other at different speeds, they meet in 1 hour

therefore, distance travelled by 1^{st} car = u km

and distance travelled by 2^{nd} car = v km

u + v =100 ....(2)

Adding both the equations, we obtain

2u = 120

u = 60

Substituting this value in equation (2), we obtain

60 + v = 100

v = 40

Equations are u - v = 20 and u + v = 100 where the speed of 1^{st} car and 2^{nd} car be u km/h and v km/h respectively.

Hence, speed of the 1^{st} car = 60 km / h and speed of the 2^{nd} car = 40 km / h

(v) Let the length and breadth of the rectangle be x unit and y unit respectively. Then the area of the rectangle is xy square units.

According to the question,

When length is reduced by 5 units and breadth is increased by 3 units, area of the rectangle gets reduced by 9 square units;

( x - 5)( y + 3) = xy - 9

xy + 3x - 5y - 15 = xy - 9

3x - 5y - 6 = 0 ....(1)

When we increase the length by 3 units and the breadth by 2 units, the area increases by 67 square units;

( x + 3)( y + 2) = xy + 67

xy + 2x + 3y + 6 = xy + 67

2x + 3y - 61 = 0 ....(2)

By cross-multiplication method, we obtain

[x/(b₁c₂- b₂c₁) = y/(c₁a₂- c₂a₁) = 1/(a₁b₂ - a₂b₁)]

x/305 - (- 108) = y/- 12 - (-183) = 1/9 - (-19)

x/323 = y/171 = 1/19

x/323 = 1/19 and y/171 = 1/19

x = 17, y = 19

Equations are 3x - 5y - 6 = 0 and 2x + 3y - 61 = 0 where length and breadth of the rectangle are x and y respectively.

Hence, the length and breadth of the given rectangle are 17 units and 9 units respectively.

**☛ Check: **NCERT Solutions for Class 10 Maths Chapter 3

**Video Solution:**

## Form the pair of linear equations in the following problems and find their solutions (if they exist) by any algebraic method:(i) A part of monthly hostel charges is fixed and the remaining depends on the number of days one has taken food in the mess. When student A takes food for 20 days, she has to pay ₹ 1000 as hostel charges whereas a student B, who takes food for 26 days, pays ₹ 1180 as hostel charges. Find the fixed charges and the cost of food per day.(ii) A fraction becomes 1/3 when 1 is subtracted from the numerator and it becomes 1/4 when 8 is added to its denominator. Find the fraction.(iii) Yash scored 40 marks in a test, getting 3 marks for each right answer and losing 1 mark for each wrong answer. Had 4 marks been awarded for each correct answer and 2 marks been deducted for each incorrect answer, then Yash would have scored 50 marks. How many questions were there in the test?(iv) Places A and B are 100 km apart on a highway. One car starts from A and another from B at the same time. If the cars travel in the same direction at different speeds, they meet in 5 hours. If they travel towards each other, they meet in 1 hour. What are the speeds of the two cars?(v) The area of a rectangle gets reduced by 9 square units, if its length is reduced by 5 units and breadth is increased by 3 units. If we increase the length by 3 units and the breadth by 2 units, the area increases by 67 square. Find the dimensions of the rectangle

NCERT Solutions for Class 10 Maths Chapter 3 Exercise 3.5 Question 4

**Summary:**

i) A part of monthly hostel charges is fixed and the remaining depends on the number of days one has taken food in the mess. When student A takes food for 20 days, she has to pay ₹ 1000 as hostel charges whereas a student B, who takes food for 26 days, pays ₹ 1180 as hostel charges. Equations are x + 20 y = 1000 and x + 26 y = 1180 where x is the fixed charge of the food and y is the charge for food per day. Hence, the fixed charge is ₹ 400 and charge per day is ₹ 30 ii) A fraction becomes 1/3 when 1 is subtracted from the numerator and it becomes 1/4 when 8 is added to its denominator. Equations are 3x - y = 3 and 4x - y = 8 where the numerator of the fraction is x, and the denominator is y. Hence, the fraction is 5/12 iii) Yash scored 40 marks in a test, getting 3 marks for each right answer and losing 1 mark for each wrong answer. Had 4 marks been awarded for each correct answer and 2 marks been deducted for each incorrect answer, then Yash would have scored 50 marks. Equations are 3x - y = 40 and 2x - y = 25 where the number of right and wrong answers are x and y respectively. Number of right answers = 15 and Number of wrong answers = 5. Hence, the total number of questions = 20 iv) Places A and B are 100 km apart on a highway. One car starts from A and another from B at the same time. If the cars travel in the same direction at different speeds, they meet in 5 hours. If they travel towards each other, they meet in 1 hour. Equations are u - v = 20 and u + v = 100 where the speed of 1^{st} car and 2^{nd} car be u km/h and v km/h respectively. Hence, speed of the 1^{st} car = 60 km / h and speed of the 2^{nd} car = 40 km / h v) The area of a rectangle gets reduced by 9 square units, if its length is reduced by 5 units and breadth is increased by 3 units. If we increase the length by 3 units and the breadth by 2 units, the area increases by 67 square. Equations are 3x - 5y - 6 = 0 and 2x + 3y - 61 = 0 where length and breadth of the rectangle are x and y respectively. Hence, the length and breadth of the rectangle are 17 units and 9 units respectively.

**☛ Related Questions:**

- Which of the following pairs of linear equations has a unique solution, no solution, or infinitely many solutions? In case there is a unique solution, find it by using the cross multiplication method. (i) x - 3y - 3 = 0; 3x - 9 y - 2 = 0 (ii) 2x + y = 5; 3x + 2 y = 8 (iii) 3x - 5 y =-20; 6x - 10 y = 40 (iv) x - 3y - 7 = 0; 3x - 3y - 15 = 0
- (i) For which values of a and b will the following pair of linear equations have an infinite number of solutions? 2x + 3y = 7 (a - b) x + (a + b) y = 3a + b - 2 (ii) For which value of k will the following pair of linear equations have no solution? 3x + y = 1 (2k -1) x + (k -1) y = 2k +1
- Solve the following pair of linear equations by the substitution and cross-multiplication methods: 8x + 5 y = 9 3x + 2 y = 4.

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