Chapter 3 – Pair Of Linear Equations In Two Variables

CONCEPT Solutions Class 10 Maths Chapter 3 – Pair Of Linear Equations In Two Variables are provided here. These solutions are prepared by our experts to help the students in their exam preparations. They solve and provide concepts, the NCERT Solutions for Maths so as to aid the students to solve the problems easily. They also focus on preparing the solutions in such a way that it is easy to understand the concepts for the students. They provide a detailed and step-wise explanation of each solution to the questions given in the exercises of NCERT books and other books.

Assertion-Reason & MCQs for Practice
Case Study Questions for Practice

A pair of Linear Equations in two variables: An equation of the form ax + by + c = 0, where a, b and c are real numbers, such that a and bare not both zero, is called a linear equation in twovariables. Two linear equations in same two variables x and y are called pair of linearequations in two variables.

    Geometrical Representation of a Linear Equation

    Geometrically, a linear equation in two variables can be represented as a straight line.

    2x – y + 1 = 0

    ⇒ y = 2x + 1

    Graph of y = 2x+1

    Plotting a Straight Line

    The graph of a linear equation in two variables is a straight line. We plot the straight line as follows:

    1. Types of Polynomials based on Degree

    Linear Polynomial

    A polynomial whose degree is one is called a linear polynomial.

    For example, 2x+1 is a linear polynomial.

    Quadratic Polynomial

    A polynomial of degree two is called a quadratic polynomial.

    For example, 3x2+8x+5 is a quadratic polynomial.

    Cubic Polynomial

    A polynomial of degree three is called a cubic polynomial.

    For example, 2x3+5x2+9x+15 is a cubic polynomial.

    • Graph of the polynomial x^n

    For a polynomial of the form y=xn where n is a whole number:

    as n increases, the graph becomes steeper or draws closer to the Y-axis

    If n is odd, the graph lies in the first and third quadrants

    If n is even, the graph lies in the first and second quadrants

    The graph of y=−xn is the reflection of the graph of y=xn on the x-axis

    1. Geometrical Meaning of Zeros of a Polynomial

    Geometrically, zeros of a polynomial are the points where its graph cuts the x-axis.

    (i) One zero(ii) Two zeros (iii) Three zeros

    Here A, B and C correspond to the zeros of the polynomial represented by the graphs.

    Number of Zeros

    In general, a polynomial of degree n has at most n zeros.

    • A linear polynomial has one zero,
    • A quadratic polynomial has at most two zeros.
    • A cubic polynomial has at most 3 zeros.
    • The general form of a pair of linear equations in two variablesis

    a1x + b1y + c1=0

    a2x + b2y + c2= 0

    where a1, a2, b1, b2, c1and c2are real numbers, such that

    • A system of linear equations in two variables represents two lines in a plane. For two given lines ina plane there could be three possiblecases:
    • The two lines are intersecting, i. e., they intersect at onepoint.
    • The two lines are parallel, i.e., they do not intersect at any realpoint.
    • The two lines are coincident lines, i.e., one line overlaps the otherline.
    • A system of simultaneous linear equations is said tobe
    • Consistent, if it has at least onesolution.
    • In-consistent, if it has nosolution.
    • If thelines
    • Intersect at a point, then that point gives the unique solution of the system of equations. In this case system of equations is said to beconsistent.
    • Coincide (overlap), then the pair of equations will have infinitely many solutions. System of equations is said to beconsistent.
    • are parallel, then the pair of equations has no solution. In this case pair of equations is said tobeinconsistent.
    • Solution of a pair of Linear Equations in twovariable:

    System of equations can be solved using Algebraic and Graphical Methods.

    • GraphicalMethod:
    • A linear equation in two variables is represented geometrically by a straightline.
    • The graph of a pair of linear equations in two variables is represented by twolines. Steps:
    • Draw the graphs of both the equations by finding two solutions foreach.
    • Plot the points and draw the lines passing through them to represent theequations.
    • The behaviour of lines representing a pair of linear equations in two variables and the existence of solutions can be summarised asfollows: