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. x – 3y – 3 = 0 3x – 9y – 2 = 0

 

 cbse ncert 10th mathematics chapter 3 exercise 3.5pair of linear equations in two variables 

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.

x – 3y – 3 = 0
3x – 9y – 2 = 0

 

a1=1 , b1 =(-3)  c1 = (-3)

a2 =3 b2 = (-9) c2 = (-2)

a1/a2 = 1/3

b1/b2 = (-3)/(-9) =1/3

c1/c2 =(-3)/(-2) =3/2

 

a1/a2 =b1/b2 =/= (not equal )c1/c2

 

system has no solution

(ii) 

2x + y = 5
3x + 2y = 8

rewrite the equations

2x+1y-5=0

3x+2y-8=0

 

a1=2 , b1 =1  c1 = (-5)

a2 =3 b2 = 2  c2 = (-8)

 

a1/a2 = 2/3

b1/b2 = 1/2

 

a1/a2 =/= (not equal) b1/b2

system has unique solution

 

use cross multiplication method

 

           x                      y              1

1                    (-5)              2               1

2                    (-8)              3              2    

x / [1(-8) - 2(-5)]  = y / [(-5)3 - (-8)2]  = 1 / [2*2 -3*1]

x /[-8+10] = y / [-15+16] = 1 / [4-3]

x/2 =y/1=1/1

using the first and last

x/2 = 1/1

x =2

using the second and last

y/1 = 1/1

y=1

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ncert cbse 10th mathematics chapter 3 optional exercise 3.7 

 The ages of two friends Ani and Biju differ by 3 years. Ani’s father  is twice as old as Ani and Biju is twice as old as his sister Cathy. The ages of Cathy and Ani’s father differ by 30 years. Find the ages of Ani and Biju

solution

 

2. One says, “Give me a hundred, friend! I shall then become twice as rich as you”. The other replies, “If you give me ten, I shall be six times as rich as you”. Tell me what is the amount of their (respective) capital? 

solution

3. A train covered a certain distance at a uniform speed. If the train would have been 10 km/h faster, it would have taken 2 hours less than the scheduled time. And, if the train were slower by 10 km/h; it would have taken 3 hours more than the scheduled time. Find the distance covered by the train. 

solution  

 

4. The students of a class are made to stand in rows. If 3 students are extra in a row, there would be 1 row less. If 3 students are less in a row, there would be 2 rows more. Find the number of students in the class.

solution  

5. In a ∆ ABC, ∠ C = 3 ∠ B = 2 (∠ A + ∠ B). Find the three angles.

solution

 

Solve the following pair of linear equations:

 px + qy = p – q 

 qx – py = p + q

solution

 

(ii) ax + by = c
     bx + ay = 1 + c

solution  

 

(iii) 

(x/a) -(y/b) = 0

ax +by = (a^2)  + (b^2)

solution  

 

(iv)

(a – b)x + (a + b) y = (a^2) – 2ab – (b^2)

(a + b)(x + y) = (a^2) + (b^2 )

solution

 

(v)

152x – 378y = – 74

–378x + 152y = – 604

solution   

 ABCD is a cyclic quadrilateral  Find the angles of the cyclic quadrilateral,

if angles are A =(4y+20) , B =(3y-5) , C=(-4x), D=(-7x+5)

solution

 

exercise 3.6

2

Ritu can row downstream 20 km in 2 hours, and upstream 4 km in 2 hours. Find her speed of rowing in still water and the speed of the current. 

solution

 

 (ii) 2 women and 5 men can together finish an embroidery work in 4 days, while 3 women and 6 men can finish it in 3 days. Find the time taken by 1 woman alone to
finish the work, and also that taken by 1 man alone.

solution

 

(iii) 

Roohi travels 300 km to her home partly by train and partly by bus. She takes 4 hours if she travels 60 km by train and the remaining by bus. If she travels 100 km by train and the remaining by bus, she takes 10 minutes longer. Find the speed of the train and the bus separately.

 

solution

 

exercise 3.5

4

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 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.

solution

 

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

 solution

(iii)Y 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?

solution 

(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?

solution 

 (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

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

8x + 5y = 9
3x + 2y = 4

solution

 

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

solution 

 

(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

 

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.

x – 3y – 3 = 0
3x – 9y – 2 = 0

solution 

(ii) 

2x + y = 5
3x + 2y = 8

solution 

 

 

 

 
 
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