#### Study Materials and Revision Notes for Ch 3 Pair of Linear Equations in two VariablesÂ Class 10th Maths

**Pair of Linear Equations in two Variables**

â€¢

__Linear equation in two variables:__Â An equation in the form of axÂ + byÂ + c = 0 where x and y are variables and a, b, c are real numbers (aâ‰ 0, bâ‰ 0) is called linear equation in two variable.

Example: (i) 3xÂ + 4yÂ + 4 = 0

Â Â Â Â Â Â Â Â (ii) 2/3 xÂ + y = 0

*â€¢Â*

__Â Two linear equations in the same two variables are called a pair of linear equations in two variables. The general form of a pair of linear equations is:__

*Pair of linear equation in two varibales:*a

_{1}x + b

_{1}y + c

_{1}Â = 0

a

_{2}x + b

_{2}y + c

_{2}Â = 0

where, a

_{1},

_{Â }b

_{1},

_{Â }c

_{1}, a

_{2},

_{Â b}

_{2},

_{Â }c

_{2 are real numbers and none of them are equal to zero.}

Example: (i) 3xÂ + 4yÂ + 6 = 0

Â Â Â Â Â Â Â Â Â Â Â xÂ + 2yÂ + 3 = 0

*â€¢Â*A pair of linear equations in two variables can be represented, and solved, by the:

(i) graphical method

(ii) algebraic method

*â€¢Â*Graphical Method :

The graph of a pair of linear equations in two variables is represented by two lines.

(i) 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.

(ii) If the lines coincide, then there are infinitely many solutions â€” each point on the

line being a solution. In this case, the pair of equations is dependent (consistent).

(iii) If the lines are parallel, then the pair of equations has no solution. In this case, the

pair of equations is inconsistent.

*â€¢*Algebraic Methods : We have discussed the following methods for finding the solution(s)

of a pair of linear equations :

(i) Substitution Method

(ii) Elimination Method

(iii) Cross-multiplication Method

*â€¢Â*There are several situations which can be mathematically represented by two equations

that are not linear to start with. But we alter them so that they are reduced to a pair of

linear equations.