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 Pdf free download is part of Class 10 Maths Notes for Quick Revision. Here we have given NCERT Class 10 Maths Notes Chapter 3 Pair of Linear equations in Two Variables. According to new CBSE Exam Pattern, MCQ Questions for Class 10 Maths Carries 20 Marks.

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 equations	Graph	Number of Solutions	Consistency of System
\(\frac { { a }_{ 1 } }{ { a }_{ 2 } } \neq \frac { { b }_{ 1 } }{ { b }_{ 2 } } \)	Intersecting lines	Unique solution	Consistent
\(\frac { { a }_{ 1 } }{ { a }_{ 2 } } =\frac { { b }_{ 1 } }{ { b }_{ 2 } } \neq \frac { c_{ 1 } }{ c_{ 2 } } \)	Parallel lines	No solution	Inconsistent
\(\frac { { a }_{ 1 } }{ { a }_{ 2 } } =\frac { { b }_{ 1 } }{ { b }_{ 2 } } =\frac { c_{ 1 } }{ c_{ 2 } } \)	Co-incident lines	Infinite solutions	Consistent

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