A pair of linear equations in two variables x and y is a1x+b1y+c1=0 a2x+b2y+c2=0 General Possibilities of Two Lines a) The lines may intersect at one point. b) The lines may never meet each other c) The lines may coincide or overlap each other. Solution of a Pair of Linear Equations There are two ways to solve a pair of linear equations: 1. Graphical method 2. Algebraic Method Graphical method We can solve a pair of linear equations by representing them graphically and observing the graph whether the lines intersect, parallel, or coincide. Consistent Pair of Linear Equations A pair of linear equations is consistent if there exists at least one solution. In such case the lines are intersecting or coincident. If the lines intersect at a point, the given pair of linear equations has a unique solution. If the lines coincide, the pair of equations has infinite number of solutions and known as dependent pair of equations. Inconsistent Pair of Linear Equations A pair of linear equations is inconsistent if there is no solution. In such case the lines are parallel. Algebraic method: There are various algebraic methods to solve a pair of linear equations: (i) Substitution Method (ii) Elimination Method (iii)Cross-multiplication Method Substitution Method a) Find the value of one convenient variable either x or y. b) Substitute this value into other equation and solve it. c) Put obtained value to any convenient equation to find the value of another variable. Elimination Method a) Express the equation in the form of ax+by = c and multiply both the equations by some suitable non-zero constants to make the coefficients of one variable (either x or y) numerically equal. b) Then add or subtract one equation from the other so that one variable gets eliminated. Solve it. c) Put obtained value to any convenient equation to find the value of another variable. Cross-Multiplication Method The solution of the pair of linear equations can be given by The pair of linear equations given by a1x+b1y+c1=0 and a2x+b2y+c2=0 is: consistent and has unique solution, if consistent (dependent) and has infinite solutions if inconsistent (no solution) if Equations Reducible to a Pair of Linear Equations in Two Variables Some pairs of equations which are not linear can be reduced to linear form by making some suitable substitutions.
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