Pair of Linear equations in Two Variables Class 10 Notes Maths Chapter 3


CBSE Class 10 Maths Notes Chapter 3 Pair of Linear equations in Two Variables

  • For any linear equation, each solution (x, y) corresponds to a point on the line. General form is given by ax + by + c = 0.
  • The graph of a linear equation is a straight line.
  • Two linear equations in the same two variables 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 and c2 are real numbers, such that a12 + b12 ≠ 0, a22 + b22 ≠ 0.
  • A pair of values of variables ‘x‘ and ‘y’ which satisfy both the equations in the given system of equations is said to be a solution of the simultaneous pair of linear equations.
  • A pair of linear equations in two variables can be represented and solved, by
    (i) Graphical method
    (ii) Algebraic method

(i) Graphical method. The graph of a pair of linear equations in two variables is presented by two lines.
(ii) Algebraic methods. Following are the methods for finding the solutions(s) of a pair of linear equations:

  1. Substitution method
  2. Elimination method
  3. Cross-multiplication method.
  • There are several situations which can be mathematically represented by two equations that are not linear to start with. But we allow them so that they are reduced to a pair of linear equations.
  • Consistent system. A system of linear equations is said to be consistent if it has at least one solution.
  • Inconsistent system. A system of linear equations is said to be inconsistent if it has no solution.

CONDITIONS FOR CONSISTENCY
Let the two equations be:
a1x + b1y + c1 = 0
a2x + b2y + c2 = 0
Then,

Relationship between coeff. or the pair of equationsGraphNumber of SolutionsConsistency of System
a1a2≠b1b2Intersecting linesUnique solutionConsistent
a1a2=b1b2≠c1c2Parallel linesNo solutionInconsistent
a1a2=b1b2=c1c2Co-incident linesInfinite solutionsConsiste

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