📝 PAIR OF LINEAR EQUATIONS IN TWO VARIABLES
● 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
`a_1x + b_1y + c_1 = 0`
`a_2x + b_2y + c_2 = 0`
where `a_1, a_2, b_1, b_2, c_1, c_2` are real numbers,
such that `a_1^2 + b1^2 ne 0, a_2^2 + b_2^2 ne 0`
● A pair of linear equations in two variables can be represented, and solved, by the:
(i) graphical method
(ii) algebraic method
(i) Graphical Method :
The graph of a pair of linear equations in two variables is represented by two lines.
(a) If the lines intersect at a point, then that point gives the unique solution of the two equations. In this case, the pair of equations is consistent.
(b) 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).
(c) If the lines are parallel, then the pair of equations has no solution. In this case, the pair of equations is inconsistent.
● If a pair of linear equations is given by `a_1x + b_1y + c_1 = 0` and `a_2x + b_2y + c_2 = 0`, then the following situations can arise :
(i) `a_1/a_2 ne b_1/b_2`: In this case, the pair of linear equations is consistent.
(ii) `a_1/a_2 = b_1/b_2 ne c_1/c_2`: In this case, the pair of linear equations is inconsistent.
(iii) `a_1/a_2 = b_1/b_2 = c_1/c_2`: In this case, the pair of linear equations is dependent and consistent.
(ii) Algebraic Methods :
(a) Substitution Method
(b) Elimination Method
(c) Cross-multiplication Method
Very interesting notes ☺
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