NCERT Solutions for Class 10 Maths Chapter 3 – Pair of Linear Equations in Two Variables Exercise 3.5

NCERT Solutions for Class 10 Maths Chapter 3 – Pair of Linear Equations in Two Variables Exercise 3.5 are designed to help students master the concepts of solving linear equations using both graphical and algebraic methods. This exercise builds on previous methods like substitution and elimination, with an emphasis on solving practical problems and real-life applications.

Exercise 3.5 provides students with problems that require them to solve word problems involving linear equations in two variables. The solutions are presented in a step-by-step format, guiding students through translating word problems into equations and then solving them using the appropriate method. The exercise is highly beneficial for developing problem-solving skills in algebra and is crucial for performing well in Class 10 exams.

These solutions are aligned with the latest CBSE syllabus, providing clear, logical explanations and ensuring students are well-prepared for exams.

NCERT Solutions for Class 10 Maths Chapter 3 – Pair of Linear Equations in Two Variables Exercise 3.5

Pair of Linear Equations in Two Variables

Medium

Q.

$\begin{array}{l}\text{Which of the following pairs of linear equations are\hspace{0.33em}consistent/inconsistent?}\\ \text{If consistent, obtain the\hspace{0.33em}solution graphically:}\\ \text{(i) \hspace{0.33em}\hspace{0.33em}x\hspace{0.33em}+\hspace{0.33em}y\hspace{0.33em}=\hspace{0.33em}5,\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}2x\hspace{0.33em}+\hspace{0.33em}2y\hspace{0.33em}=\hspace{0.33em}10 \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\\ \text{(ii)\hspace{0.33em}\hspace{0.33em}x\hspace{0.33em}–\hspace{0.33em}y\hspace{0.33em}=\hspace{0.33em}8,\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.17em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}3x\hspace{0.33em}–\hspace{0.33em}3y\hspace{0.33em}=\hspace{0.33em}16}\\ \text{(iii)\hspace{0.33em}2x\hspace{0.33em}+\hspace{0.33em}y\hspace{0.33em}}–\text{6\hspace{0.33em}=\hspace{0.33em}0,\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}4x\hspace{0.33em}–\hspace{0.33em}2y\hspace{0.33em}}–\text{\hspace{0.33em}4\hspace{0.33em}=\hspace{0.33em}0}\\ \text{(iv)\hspace{0.33em}2x\hspace{0.33em}–\hspace{0.33em}2y\hspace{0.33em}–\hspace{0.33em}2\hspace{0.33em}=\hspace{0.17em}0,\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}4x\hspace{0.33em}–\hspace{0.33em}4y\hspace{0.33em}}–\text{\hspace{0.33em}5\hspace{0.33em}=\hspace{0.33em}0\hspace{0.17em}\hspace{0.17em}}\end{array}$

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Pair of Linear Equations in Two Variables

Medium

Q.

$\begin{array}{l}\text{(i) For which values of a and\hspace{0.33em}b does the following pair\hspace{0.33em}of linear equations have an }\\ \text{\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}infinite number of\hspace{0.33em}solutions?}\\ \text{ \hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}2x\hspace{0.33em}+\hspace{0.33em}3y\hspace{0.33em}=7}\\ \text{ \hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}(a\hspace{0.33em}–\hspace{0.33em}b)x\hspace{0.33em}+\hspace{0.33em}(a\hspace{0.33em}+\hspace{0.33em}b)y\hspace{0.33em}=\hspace{0.33em}3a\hspace{0.33em}+\hspace{0.33em}b\hspace{0.33em}}–\text{\hspace{0.33em}2}\\ \text{(ii) For which value of k will the following pair\hspace{0.33em}of linear equations have no solution?}\\ \text{\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}3x\hspace{0.33em}+\hspace{0.33em}y\hspace{0.33em}=\hspace{0.33em}1}\\ \text{\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}(2k\hspace{0.33em}–\hspace{0.33em}1)x\hspace{0.33em}+\hspace{0.33em}(k\hspace{0.33em}–\hspace{0.33em}1)\hspace{0.33em}y\hspace{0.33em}=\hspace{0.33em}2k\hspace{0.33em}+\hspace{0.33em}1}\end{array}$

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Pair of Linear Equations in Two Variables

Medium

Q.

$\begin{array}{l}\text{Solve the following pair of linear equations by the}\\ \text{substitution and cross-multiplication methods:}\\ 8\mathrm{x}+5\mathrm{y}=9\\ 3\mathrm{x}+2\mathrm{y}=4\end{array}$

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Pair of Linear Equations in Two Variables

Medium

Q.

$\begin{array}{l}\text{Form the pair of linear equations in the following}\\ \text{problems and find their solutions (if they exist)}\\ \text{by any algebraic method:}\\ \text{ (i) A part of monthly hostel charges is fixed and the}\\ \text{ remaining depends on the number of days one}\\ \text{ has taken food in the mess. When a student A}\\ \text{ takes food for 20 days she has to pay }₹\text{1000}\\ \text{ as hostel charges whereas a student B, who}\\ \text{ takes food for 26 days, pays }₹\text{1180 as hostel}\\ \text{ charges. Find the fixed charges and the cost of }\\ \text{ food per day.}\\ \text{(ii) A fraction becomes }\frac{1}{3}\text{ when 1 is subtracted from }\\ \text{ the numerator and it becomes }\frac{1}{4}\text{ when 8 is added}\\ \text{ to its denominator. Find the fraction.}\\ \text{(iii) Yash scored 40 marks in a test, getting 3 marks}\\ \text{ for each right answer and losing 1 mark for each}\\ \text{ wrong answer. Had 4 marks been awarded for}\\ \text{ each correct answer and 2 marks been deducted}\\ \text{ for each incorrect answer, then Yash would have}\\ \text{ scored 50 marks. How many questions were there}\\ \text{ in the test?}\end{array}$
$\begin{array}{l}\text{(iv) Places A and B are 100 km apart on a highway. One}\\ \text{ car starts from A and another from B at the same}\\ \text{ time. If the cars travel in the same direction at}\\ \text{ different speeds, they meet in 5 hours. If they}\\ \text{ travel towards each other, they meet in 1 hour.}\\ \text{ What are the speeds of the two cars?}\\ \text{(v) The area of a rectangle gets reduced by 9 square}\\ \text{ units, if its length is reduced by 5 units and}\\ \text{ breadth is increased by 3 units. If we increase}\\ \text{ the length by 3 units and the breadth by 2 units},\\ \text{ the area increases by 67 square units}.\text{ Find the}\\ \text{ dimensions of the rectangle.}\end{array}$

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Q1: Which of the following pairs of linear equations are consistent/inconsistent? If consistent, obtain the solution graphically:

1. x + y = 5,
   2x + 2y = 10

2. x - y = 8,
   3x - 3y = 16

3. 2x + y - 6 = 0,
   4x - 2y - 4 = 0

4. 2x - 2y - 2 = 0,
   4x - 4y - 5 = 0

Solution:

• Solve the system of equations to check if the equations are consistent or inconsistent.

• If consistent, graphically find the point of intersection of the lines represented by the equations.

View Solution for Q1

Q2: For which values of a and b does the following pair of linear equations have an infinite number of solutions?

1. 2x + 3y = 7,
   (a - b)x + (a + b)y = 3a + b - 2

2. 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:

• Solve for values of a and b that will make the system have infinite solutions.

• Similarly, find the value of k for which the system has no solution.

View Solution for Q2

Q3: Solve the following pair of linear equations by the substitution and cross-multiplication methods:

1. 8x + 5y = 9
   3x + 2y = 4

Solution:

• Use the substitution method to express one variable in terms of the other and substitute into the second equation.

• Alternatively, apply the cross-multiplication method to solve the system of equations.

View Solution for Q3

Q4: Form the pair of linear equations in the following problems and find their solutions (if they exist) by any algebraic method:

1. 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 ₹1000 as hostel charges, whereas student B, who takes food for 26 days, pays ₹1180. Find the fixed charges and the cost of food per day.

2. 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.

3. 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?

4. 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, they meet in 5 hours. If they travel towards each other, they meet in 1 hour. What are the speeds of the two cars?

5. The area of a rectangle gets reduced by 9 square units if its length is reduced by 5 units and its 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:

• For each word problem, form the pair of linear equations based on the given conditions and solve them algebraically.

View Solution for Q4

FAQs: Class 10 Maths Chapter 3 – Exercise 3.5

Q1. What is the main focus of Exercise 3.5?
Answer:
Exercise 3.5 focuses on solving word problems related to pair of linear equations in two variables using algebraic methods (substitution and elimination), and applying them to real-life situations.

Q2. How do I approach solving word problems in this exercise?
Answer:
Follow these steps:

1. Read the problem carefully and identify the unknowns (variables).

2. Translate the problem into a pair of linear equations in two variables.

3. Solve the equations using either the substitution method or the elimination method.

4. Check your answer by substituting it back into the original equations.

Q3. What are the key skills needed to solve word problems in this exercise?
Answer:
You need to:

• Translate real-world situations into algebraic equations.

• Identify the correct method to solve the equations (substitution or elimination).

• Interpret the solutions in the context of the word problem.

Q4. How do I ensure accuracy while solving word problems in Exercise 3.5?
Answer:

• Break the problem down into manageable parts.

• Write the equations clearly before solving.

• Double-check your calculations for accuracy.

Q5. How do NCERT Solutions help with exam preparation?
Answer:
These solutions provide detailed, step-by-step solutions, helping students understand how to apply algebraic methods to word problems. By practicing these solutions, students can improve their conceptual clarity and develop problem-solving skills, ensuring better performance in Class 10 exams.