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NCERT Class 10 Maths Solutions Chapter - 3 Pairs of Linear Equations in Two Variables, Ex-3.5

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.



Solution


   Thus, the given pair of equations has no solution.
 Thus, the given pair of equations has unique solution.

              By cross-multiplication method,



 Thus, the given pair of equations has infinite solutions.
             The solutions can be obtained by assuming the value of x to be k, where k is any  
             constant. So, the ordered pairs , where k is a constant, are the solutions of
             the given pair of equations.
 Thus, the given pair of equations has unique solution.

             By cross-multiplication,
Concept Insight: In order to answer such questions, remember the condition for the pair of linear equations to have unique, infinitely many or no solution relating the coefficients. Also, remember the formula used to solve by cross multiplication method. While applying the formula, be careful about the signs of the coefficients. And above all don't forget to first write the linear equations in standard form which is ax + by + c = 0.

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?

Solution

Concept insight: In order to solve such problems, firstly write the linear equations in standard form which is ax + by + c = 0.To find the value of the unknowns, the key idea is to remember the conditions for a given pair of equations to have infinite solutions and no solution.  In case of infinite solutions rule out the values which does not satisfy all the ratios. Also, be careful about the signs of the coefficients.

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

Solution

Substituting this value of y in equation (1), we obtain:
Cross-multiplication method:
Concept insight: In order to solve the given pair of equations by cross multiplication method, remember the formula to be used and convert the system of equations to standard form. While applying the formula, be careful about the signs of the coefficients. 

In order to solve the given pair of equations by substitution method, substitute the value of any one of the variable from any one of the equation. Make sure you substitute the value of that variable which simplifies your calculations. 
Solution will be same in both cases.

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 when 1 is subtracted from the numerator and it becomes 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 the fixed charge of the food and the charge for food per day be x and y respectively.
According to the question,
x + 20y = 1000     ...(1) 
x + 26y = 1180   ...(2) 

Substituting this value of y in equation (1) from equation (2), we obtain 
6y = 180 
y = 30 

Substituting this value of y in equation (1), we obtain:
x+ 20 x 30 = 1000 
x = 1000 - 600 
x = 400 

Thus, the fixed charge of the food and the charge per day are Rs 400 and Rs 30 respectively.

Concept insight: Here, the fixed charge of the food and charge for food per day are the unknown quantities. So they are taken as variables x and y. The two equations can then be obtained by using the given conditions. You will observe that the variable x has the same coefficient in both the equations, so it will be easier to find the solution by eliminating x from both the equations. Also, one can solve the system by other methods. 

(ii)  Let the fraction be  According to the question, 

According to the question,
Subtracting equation (1) from equation (2), we obtain: 
x = 5
Putting the value of x in equation (1), we obtain:
15 - y = 3 
y = 12
Concept insight: Since the problem asks for a fraction. The numerator and denominator of the fraction need to be represented by variables. A pair of linear equations can be obtained from the given conditions. Observe that the variable y has the same coefficient in both the equations, so it will be easier to find the solution by eliminating y from both the equations. 

(iii)  Let the number of right answers and wrong answers be x and y 
respectively.


According to the question,
Subtracting equation (2) from equation (1), we obtain:
x = 15
Substituting the value of x in equation (2), we obtain:
30 - y = 25
y = 5

Thus, the number of right answers and the number of wrong answers is 15 and 5 respectively.
Therefore, the total number of questions is 20. 
Concept insight: In this problem, the number of write answers and the number of wrong answers answered by Yash are the unknown quantities. So, they must be represented by two variables. A pair of linear  equations can be obtained by applying the given condition. Variable y has the same coefficient in both the equations, so it will be easier to find the solution by eliminating y from both the equations. 

(iv)  Let the speed of first car and second car be u km/h and v km/h respectively.

Speed of both cars while they are travelling in same direction = (u - v) km/h
Speed of both cars while they are travelling in opposite directions i.e., when they are travelling towards each other = (u + v) km/h

Distance travelled = Speed x Time

According to the question,
Substituting the value of u in equation (2), we obtain:
v = 40

Hence, speed of the first car is 60 km/h and speed of the second car is 40 km/h.

Concept insight: In this problem, the speeds of the two cars are not known so we will represent them by variables. Remember that when the cars are travel in the same direction, then the speed will be equal to the difference of their speeds and when they travel in opposite direction, then the speed will be equal to the sum of their speeds. Don't miss to write the unit.

(v)  Let length and breadth of rectangle be x unit and y unit respectively. 

Area = xy

According to the question,
Thus, the length and breadth of the rectangle are 17 units and 9 units respectively.

Concept insight: Here, the length and the breadth of the rectangle will be represented by variables. Then, the pair of equations will be written from the given conditions. Solution can be easily computed  using the cross multiplication method.

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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.



Solution


   Thus, the given pair of equations has no solution.
 Thus, the given pair of equations has unique solution.

              By cross-multiplication method,



 Thus, the given pair of equations has infinite solutions.
             The solutions can be obtained by assuming the value of x to be k, where k is any  
             constant. So, the ordered pairs , where k is a constant, are the solutions of
             the given pair of equations.
 Thus, the given pair of equations has unique solution.

             By cross-multiplication,
Concept Insight: In order to answer such questions, remember the condition for the pair of linear equations to have unique, infinitely many or no solution relating the coefficients. Also, remember the formula used to solve by cross multiplication method. While applying the formula, be careful about the signs of the coefficients. And above all don't forget to first write the linear equations in standard form which is ax + by + c = 0.

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?

Solution

Concept insight: In order to solve such problems, firstly write the linear equations in standard form which is ax + by + c = 0.To find the value of the unknowns, the key idea is to remember the conditions for a given pair of equations to have infinite solutions and no solution.  In case of infinite solutions rule out the values which does not satisfy all the ratios. Also, be careful about the signs of the coefficients.

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

Solution

Substituting this value of y in equation (1), we obtain:
Cross-multiplication method:
Concept insight: In order to solve the given pair of equations by cross multiplication method, remember the formula to be used and convert the system of equations to standard form. While applying the formula, be careful about the signs of the coefficients. 

In order to solve the given pair of equations by substitution method, substitute the value of any one of the variable from any one of the equation. Make sure you substitute the value of that variable which simplifies your calculations. 
Solution will be same in both cases.

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 when 1 is subtracted from the numerator and it becomes 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 the fixed charge of the food and the charge for food per day be x and y respectively.
According to the question,
x + 20y = 1000     ...(1) 
x + 26y = 1180   ...(2) 

Substituting this value of y in equation (1) from equation (2), we obtain 
6y = 180 
y = 30 

Substituting this value of y in equation (1), we obtain:
x+ 20 x 30 = 1000 
x = 1000 - 600 
x = 400 

Thus, the fixed charge of the food and the charge per day are Rs 400 and Rs 30 respectively.

Concept insight: Here, the fixed charge of the food and charge for food per day are the unknown quantities. So they are taken as variables x and y. The two equations can then be obtained by using the given conditions. You will observe that the variable x has the same coefficient in both the equations, so it will be easier to find the solution by eliminating x from both the equations. Also, one can solve the system by other methods. 

(ii)  Let the fraction be  According to the question, 

According to the question,
Subtracting equation (1) from equation (2), we obtain: 
x = 5
Putting the value of x in equation (1), we obtain:
15 - y = 3 
y = 12
Concept insight: Since the problem asks for a fraction. The numerator and denominator of the fraction need to be represented by variables. A pair of linear equations can be obtained from the given conditions. Observe that the variable y has the same coefficient in both the equations, so it will be easier to find the solution by eliminating y from both the equations. 

(iii)  Let the number of right answers and wrong answers be x and y 
respectively.


According to the question,
Subtracting equation (2) from equation (1), we obtain:
x = 15
Substituting the value of x in equation (2), we obtain:
30 - y = 25
y = 5

Thus, the number of right answers and the number of wrong answers is 15 and 5 respectively.
Therefore, the total number of questions is 20. 
Concept insight: In this problem, the number of write answers and the number of wrong answers answered by Yash are the unknown quantities. So, they must be represented by two variables. A pair of linear  equations can be obtained by applying the given condition. Variable y has the same coefficient in both the equations, so it will be easier to find the solution by eliminating y from both the equations. 

(iv)  Let the speed of first car and second car be u km/h and v km/h respectively.

Speed of both cars while they are travelling in same direction = (u - v) km/h
Speed of both cars while they are travelling in opposite directions i.e., when they are travelling towards each other = (u + v) km/h

Distance travelled = Speed x Time

According to the question,
Substituting the value of u in equation (2), we obtain:
v = 40

Hence, speed of the first car is 60 km/h and speed of the second car is 40 km/h.

Concept insight: In this problem, the speeds of the two cars are not known so we will represent them by variables. Remember that when the cars are travel in the same direction, then the speed will be equal to the difference of their speeds and when they travel in opposite direction, then the speed will be equal to the sum of their speeds. Don't miss to write the unit.

(v)  Let length and breadth of rectangle be x unit and y unit respectively. 

Area = xy

According to the question,
Thus, the length and breadth of the rectangle are 17 units and 9 units respectively.

Concept insight: Here, the length and the breadth of the rectangle will be represented by variables. Then, the pair of equations will be written from the given conditions. Solution can be easily computed  using the cross multiplication method.

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