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

NCERT solutions for class 10 Maths Chapter 3 – Pair of Linear Equations in Two Variables provide the detailed answers for questions given in CBSE NCERT 10 Maths textbook. CBSE NCERT Class 10 Maths Chapter 3 tells you about algebraic and graphical methods for solving linear equations. This chapter contains 7 exercises, consisting of a total of 29 questions. Our comprehensive  NCERT Solutions enable you to gain mastery over the algebraic and graphical methods for solving linear equations.

Pair of Linear Equations in Two Variables are designed in a logical and explanatory way by a team of maths experts. They follow the latest syllabus given by CBSE. Our NCERT Solutions help you become thorough with all concepts of the chapters and help you build a strong conceptual base.

Maths Chapter 3 – Pair of Linear Equations in Two Variables are written very meticulously and are very useful for revision during class 10 board exams. They cover all the question types that are important from an exam point of view. It helps you to take the board exams with confidence and score good marks.

NCERT Class 10 Maths Chapter 3 – Pair of Linear Equations in Two Variables

 

  • Introduction

 

This portion of NCERT CBSE Class 10 Maths Chapter 3 introduces you to a pair of linear equations derived from a daily life situation and tells you that this chapter teaches you the ways to solve such a pair of equations. 

 

  • Pair of Linear Equations in Two Variables

 

Here you will learn that when there are two linear equations in the same two variables x and y, they are called a pair of linear equations in two variables.

The most general form of a pair of linear equations is 

a1x + b1y + c1 = 0 

a2x + b2y + c2 = 0 where a1, a2, b1, b2, c1, c2 are real numbers, such that

a12 + b12 ≠ 0; a22 + b22 ≠ 0

Every linear equation can be represented geometrically (graphically) also to give a straight line and every solution of the equation is a point on the line representing it. 

For any linear equation, each solution (x, y) of a linear equation in two variables, 

ax + by + c = 0, corresponds to a point on the line representing the equation, and vice versa.

 

  • Graphical Method of Solution of a Pair of Linear Equations

 

This section of the chapter will teach you that a pair of linear equations in two variables will be two straight lines, both to be considered together with one of the following possibilities –

 (i) If the lines intersect at a point, then that point gives the unique solution (i.e. one and only one solution for this pair of linear equations in two variables) of the two

equations. In this case, the pair of equations is consistent.

(ii) If the lines coincide, then there are infinitely many solutions — each point on the

line being a solution. In this case, the pair of equations is dependent (consistent).

(iii) If the lines are parallel, then the pair of equations has no solution. In this case, the pair of equations is inconsistent.

 

  • Algebraic Methods of Solving a Pair of Linear Equations

 

This section of CBSE NCERT Solutions for Class 10 Maths Chapter 3 teaches you that the graphical method is not convenient in cases when the point representing the solution of the linear equations has non-integral coordinates, so the alternative is algebraic methods such as the following

  1. Substitution method – Here value of one variable is substituted by expressing it in terms of the other variable to solve the pair of linear equations. 
  2. Elimination Method – This method uses elimination or removal of one variable to get a linear equation in one variable. The equation is solved to get the value of the variable left, and the obtained value is substituted in either of the original equation to get the value of the variable eliminated earlier. 
  3. Cross – Multiplication Method – For the pair of linear equations a1+ b1+ c1 = 0 and a2x + b2y + c2=0, x and y can be calculated as

x = (b1c2−b2c1)/(a1b2−a2b1)

y = (c1a2−c2a1)/(a1b2−a2b1)

If a pair of linear equations is given by a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0, then the following situations can arise: (i) a1/a2 ≠ b1/b2: In this case, the pair of linear equations is consistent. (ii) a1/a2 = b1/b2 ≠ c1/c2: In this case, the pair of linear equations is inconsistent. (iii) a1/a2 = b1/b2 = c1/c2: In this case, the pair of linear equations is dependent and consistent

 

  • Equations Reducible to a Pair of Linear Equations in Two Variables

 

This part of the chapter tells you how to solve those pairs of equations which are not linear. Such pairs of equations can be reduced to linear form by making some suitable substitutions and then the pair of equations could be solved. After solving, they can be back substituted to get the values of x and y (the initial variables).

You can refer to our NCERT solutions for class 10 Maths Chapter 3 – Pair of Linear Equations in Two Variables to have an adequate practice of all the above concepts.

NCERT Class 10 Maths Chapter 3 – Pair of Linear Equations in Two Variables Exercises

NCERT solutions for class 10 Maths Chapter 3 – Pair of Linear Equations in Two Variables are framed with utmost care to explain all the concepts and answer all questions of all the 7 exercises of this chapter. The 7 exercises comprise a total of 29 questions. These solutions will surely make you comfortable and fluent in solving problems related to linear equations in two variables of CBSE NCERT Class 10 Maths Chapter 3.

Exercise 3.1 

This exercise consists of 3 questions. All the three questions of the first exercise given in Class 10 Maths Pair of Linear Equations in Two Variables ask you to convert the given information to a pair of linear equations and represent them geometrically and algebraically.

Exercise 3.2 

This exercise comprises 7 questions. Questions 1 and 7 ask you to find a graphical solution of equations/situations given. Question 2 and 3 ask you to find whether the lines representing the given pairs of linear equations intersect at a point, are parallel or coincident; consistent or inconsistent. Question 4 asks you to find whether the lines representing the given pairs of linear equations are consistent or inconsistent and find solutions in the former case. Questions 5 and 6 ask you to write a linear equation and solve.

Exercise 3.3 – All the three questions of the third exercise given in Class 10 Maths Pair of Linear Equations in Two Variables ask you to either solve the given pair of linear equation using substitution method or convert the given information to a pair of linear equations first and then solve using the substitution method.

Exercise 3.4

This exercise has 2 questions. Question no. 1 asks you to solve the given pair of linear equations by the elimination method and the substitution method. Question 2 asks you to form the pair of linear equations in the given problems and find their solutions by the elimination method.

Exercise 3.5

This exercise has 4 questions. Questions 1 and 2 ask you to check the given pairs of linear equations for the availability of a unique solution, no solution, or infinitely many solutions. Questions 3 and 4 ask you to use algebraic methods to solve the equations. 

Exercise 3.6 

This exercise has 2 questions. Question 1 asks you to solve the given pairs of equations by reducing them to a pair of linear equations. Question 2 asks you to first prepare the equations and then solve.

Exercise 3.7

This exercise in NCERT CBSE Class 10 Maths Chapter 3 has a total of 8 questions based on the direct and indirect application of all the concepts learnt in the chapter to solve them. 

NCERT Solutions for Class 10 Maths Chapter 3 – Pair of Linear Equations in Two Variables are smart and effective solutions to all the questions asked in the above exercises.