### Chapter 3 Pair of Linear Equations in Two Variables Ex 3.5

**Question 1.**

Which of the following pairs of linear equations has unique solution, no solution, or infinitely many solutions. In case there is a unique solution, find it by using cross multiplication method. **(i)** x – 3y – 3 = 0

3x – 9y – 2 = 0 **(ii)** 2x + y = 5

3x + 2y = 8 **(iii)** 3x – Sy = 20

6x – 10y = 40 **(iv)** x – 3y – 7 = 0

3x – 3y – 15 = 0 **Solution:**

**(iii)** Equations are 3x – 5y = 20 and 6x – 10y = 40

Here,

**(iv) **Equations are x – 3y = 7 and 3x – 3y = 15

**Question 2.** **(i)** for which values of a and b does the following pai of linea equation 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 equation have no solution₹

3x + y = 1

(2k – 1)x + (k – 1)y = 2k + 1 **Solution:** **(i)** Equations are

⇒ a – 2b = 3 Solving (iii) and (iv) for a and b **By cross multiplication method.**

**Question 3.**

Solve the following pair of linear equations by the substitution and cross-multiplication methods:

8x + 5y = 9

3x + 2y = 4 **Solution:**

Equations are

**By coss multiplication Method :** Equations are

**Question 4.**

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 a 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 units. Find the dimensions of the rectangle. **Solution:**

(i) Let fixed monthly hostel charges = and charges per day = ₹ y

A.T.Q.

As per condition of student A

x + 20y = 1000

As per condition of student B

x + 26y = 1180

By cross multiplication method

∴ Fixed monthly hostel charges = ₹ 400 and charges per day = ₹ 30

By cross multiplication method

Solving (i) and (ii) for x and y

By cross multiplication method

By cross multiplication method