NCERT Solutions, Exercise 3.5, Linear Equation in Two Variables, Class 10 (Mathematics) PDF Download

 Page 1


Mathematics 
(www.tiwariacademy.com: Focus on free education) 
(Chapter – 3) (Linear equations in two variables)  
(Class – X) 
 
 www.tiwariacademy.com 
1 
Exercise 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.  
  
 
Answer 1:  
  
 
Therefore, the given sets of lines are parallel to each other. Therefore, they will not 
intersect each other and thus, there will not be any solution for these equations.  
 
  
 
Therefore, they will intersect each other at a unique point and thus, there will be a unique 
solution for these equations. By cross-multiplication method,  
Page 2


Mathematics 
(www.tiwariacademy.com: Focus on free education) 
(Chapter – 3) (Linear equations in two variables)  
(Class – X) 
 
 www.tiwariacademy.com 
1 
Exercise 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.  
  
 
Answer 1:  
  
 
Therefore, the given sets of lines are parallel to each other. Therefore, they will not 
intersect each other and thus, there will not be any solution for these equations.  
 
  
 
Therefore, they will intersect each other at a unique point and thus, there will be a unique 
solution for these equations. By cross-multiplication method,  
Mathematics 
(www.tiwariacademy.com: Focus on free education) 
(Chapter – 3) (Linear equations in two variables)  
(Class – X) 
 
 www.tiwariacademy.com 
2 
  
? x = 2, y = 1  
 
  
 
Therefore, the given sets of lines will be overlapping each other i.e., the lines will be 
coincident to each other and thus, there are infinite solutions possible for these 
equations.  
 
  
Therefore, they will intersect each other at a unique point and thus, there will be a unique 
solution for these equations.  
By cross-multiplication,  
Page 3


Mathematics 
(www.tiwariacademy.com: Focus on free education) 
(Chapter – 3) (Linear equations in two variables)  
(Class – X) 
 
 www.tiwariacademy.com 
1 
Exercise 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.  
  
 
Answer 1:  
  
 
Therefore, the given sets of lines are parallel to each other. Therefore, they will not 
intersect each other and thus, there will not be any solution for these equations.  
 
  
 
Therefore, they will intersect each other at a unique point and thus, there will be a unique 
solution for these equations. By cross-multiplication method,  
Mathematics 
(www.tiwariacademy.com: Focus on free education) 
(Chapter – 3) (Linear equations in two variables)  
(Class – X) 
 
 www.tiwariacademy.com 
2 
  
? x = 2, y = 1  
 
  
 
Therefore, the given sets of lines will be overlapping each other i.e., the lines will be 
coincident to each other and thus, there are infinite solutions possible for these 
equations.  
 
  
Therefore, they will intersect each other at a unique point and thus, there will be a unique 
solution for these equations.  
By cross-multiplication,  
Mathematics 
(www.tiwariacademy.com: Focus on free education) 
(Chapter – 3) (Linear equations in two variables)  
(Class – X) 
 
 www.tiwariacademy.com 
3 
  
 
Question 2:  
(i) For which values of a and b will the following pair of linear equations have an infinite 
number of solutions?  
  
(ii) For which value of k will the following pair of linear equations have no solution?  
  
 
Answer 2:  
  
For infinitely many solutions,  
  
Subtracting (1) from (2), we obtain  
?     
Page 4


Mathematics 
(www.tiwariacademy.com: Focus on free education) 
(Chapter – 3) (Linear equations in two variables)  
(Class – X) 
 
 www.tiwariacademy.com 
1 
Exercise 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.  
  
 
Answer 1:  
  
 
Therefore, the given sets of lines are parallel to each other. Therefore, they will not 
intersect each other and thus, there will not be any solution for these equations.  
 
  
 
Therefore, they will intersect each other at a unique point and thus, there will be a unique 
solution for these equations. By cross-multiplication method,  
Mathematics 
(www.tiwariacademy.com: Focus on free education) 
(Chapter – 3) (Linear equations in two variables)  
(Class – X) 
 
 www.tiwariacademy.com 
2 
  
? x = 2, y = 1  
 
  
 
Therefore, the given sets of lines will be overlapping each other i.e., the lines will be 
coincident to each other and thus, there are infinite solutions possible for these 
equations.  
 
  
Therefore, they will intersect each other at a unique point and thus, there will be a unique 
solution for these equations.  
By cross-multiplication,  
Mathematics 
(www.tiwariacademy.com: Focus on free education) 
(Chapter – 3) (Linear equations in two variables)  
(Class – X) 
 
 www.tiwariacademy.com 
3 
  
 
Question 2:  
(i) For which values of a and b will the following pair of linear equations have an infinite 
number of solutions?  
  
(ii) For which value of k will the following pair of linear equations have no solution?  
  
 
Answer 2:  
  
For infinitely many solutions,  
  
Subtracting (1) from (2), we obtain  
?     
Mathematics 
(www.tiwariacademy.com: Focus on free education) 
(Chapter – 3) (Linear equations in two variables)  
(Class – X) 
 
 www.tiwariacademy.com 
4 
  
Substituting this in equation (2), we obtain  
  
Hence, a = 5 and b = 1 are the values for which the given equations give infinitely 
many solutions.  
  
For no solution,  
  
  
Hence, for k = 2, the given equation has no solution.  
 
Question 3:  
Solve the following pair of linear equations by the substitution and cross multiplication 
methods:  
  
 
Answer 3:  
  
From equation (ii), we obtain  
  
Substituting this value in equation (i), we obtain  
Page 5


Mathematics 
(www.tiwariacademy.com: Focus on free education) 
(Chapter – 3) (Linear equations in two variables)  
(Class – X) 
 
 www.tiwariacademy.com 
1 
Exercise 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.  
  
 
Answer 1:  
  
 
Therefore, the given sets of lines are parallel to each other. Therefore, they will not 
intersect each other and thus, there will not be any solution for these equations.  
 
  
 
Therefore, they will intersect each other at a unique point and thus, there will be a unique 
solution for these equations. By cross-multiplication method,  
Mathematics 
(www.tiwariacademy.com: Focus on free education) 
(Chapter – 3) (Linear equations in two variables)  
(Class – X) 
 
 www.tiwariacademy.com 
2 
  
? x = 2, y = 1  
 
  
 
Therefore, the given sets of lines will be overlapping each other i.e., the lines will be 
coincident to each other and thus, there are infinite solutions possible for these 
equations.  
 
  
Therefore, they will intersect each other at a unique point and thus, there will be a unique 
solution for these equations.  
By cross-multiplication,  
Mathematics 
(www.tiwariacademy.com: Focus on free education) 
(Chapter – 3) (Linear equations in two variables)  
(Class – X) 
 
 www.tiwariacademy.com 
3 
  
 
Question 2:  
(i) For which values of a and b will the following pair of linear equations have an infinite 
number of solutions?  
  
(ii) For which value of k will the following pair of linear equations have no solution?  
  
 
Answer 2:  
  
For infinitely many solutions,  
  
Subtracting (1) from (2), we obtain  
?     
Mathematics 
(www.tiwariacademy.com: Focus on free education) 
(Chapter – 3) (Linear equations in two variables)  
(Class – X) 
 
 www.tiwariacademy.com 
4 
  
Substituting this in equation (2), we obtain  
  
Hence, a = 5 and b = 1 are the values for which the given equations give infinitely 
many solutions.  
  
For no solution,  
  
  
Hence, for k = 2, the given equation has no solution.  
 
Question 3:  
Solve the following pair of linear equations by the substitution and cross multiplication 
methods:  
  
 
Answer 3:  
  
From equation (ii), we obtain  
  
Substituting this value in equation (i), we obtain  
Mathematics 
(www.tiwariacademy.com: Focus on free education) 
(Chapter – 3) (Linear equations in two variables)  
(Class – X) 
 
 www.tiwariacademy.com 
5 
  
Substituting this value in equation (ii), we obtain  
  
Again, by cross-multiplication method, we obtain  
  
 
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 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 1/3 when 1 is subtracted from the numerator and it becomes 
¼ 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?  
  
Hence,