NCERT SOLUTIONS FOR CLASS 10 MATHS PAIR OF LINEAR EQUATIONS IN TWO VARIABLES CHAPTER 3 EX 3.5
Question 1.
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 – 9y – 2 = 0
(ii) 2x + y = 5, 3x + 2y = 8
(iii) 3x – 5y = 20, 6x – 10y = 40
(iv) x – 3y – 7 = 0, 3x – 3y – 15 = 0
Solution:
Question 2.
(i) For which values of a and b does 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)
Solution:
Question 3.
Solve the following pair of linear equations by the substitution and cross-multiplication methods:
8x + 5y = 9, 3x + 2y = 4
Solution:
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 student A takes food for 20 days she has to pay Rs 1000 as hostel charges, whereas a student B, who takes food for 26 days, pays Rs 1180 as hostel charges. Find the fixed charges and the cost of food per day.
(ii) A fraction becomes 13 when 1 is subtracted from the numerator and it becomes 14 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 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 = Rs x
Charges per day = Rs y
According to Question
As per the condition of the student A
x + 20y = 1000
As per the condition of the student B
x + 26y = 1180
By Cross multiplication method
Fixed monthly hostel charges = Rs 400 and Charges per day = Rs 30
(ii) Let numerator = x and denominator = y
Fraction = x / y
According to Question
1st condition:
Equation (i)
2nd condition:
Equation (ii)
Solving Equation (i) and (ii) for x and y
By Cross multiplication method
(iii) Let number of right answers = x
Number of wrong answers = y
Total questions are x + y
According to Question
1st condition:
Equation (i)
3x – y = 40
2nd condition:
Equation (ii)
4x – 2y = 50
Solving equation (i) and (ii) for x and y
By Cross multiplication method
(iv) Let speed of car starting from point A = x km/h
Speed of car starting from point B = y km/h
According to Question
1st condition: If the cars travel in same direction, then
Equation (i)
5x – 5y = 100
or, x – y = 20
2nd condition: If they travel in opposite directions, then
Equation (ii)
x + y = 100
Solving equation (i) and (ii) for x and y
By Cross multiplication method
Speed of car starting from point A = 60 km/h.
Speed of car starting from point B = 40 km/h.