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

### Pair of Linear Equations in Two Variables Class 10 Notes Understanding the Lesson

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 equation is

a_{1}x + b_{1}y + c_{1} = 0

a_{2}x+ b_{2}y + c_{2} = 0

where a_{1},a_{2}, b_{1,}b_{2}, c_{1} c_{2} are real numbers. For the pair of linear equations, the following situations can arise:

(i) \(\frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}}\) In this case, the pair of linear equations is consistent.

(ii) \(\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}}\) The pair of linear equations in inconsistent.

(iii) \(\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}\) The pair of linear equations is consistent.

^{
}2. A pair of linear equations in two variables, can be represented, and solved by the

- Graphical method
- Algebraic method

3. Graphical Method: The graph of a pair of linear equations in two variables is represented by two lines, following three possibilities can occur.

- Two lines intersect at one point, then that point gives the unique solution of the two equations and the pair of equations is consistent.
- Two lines will not intersect, i.e. they are parallel, the pair of linear equations is inconsistent and the pair of equations will have no solution.

- The graph will be a pair of coincident lines. Each point on the lines will be a solution, so the pair of equations will have infinitely many solution and is consistent.

4. Algebraic Method: A pair of linear equations can be solved by any of the following three methods:

- Substitution method
- Elimination method
- Cross-multiplication method

5. Graphical Method of Solution of a pair of Linear Equations:

If the lines represented by the pair of linear equations in two variables are given by

a_{1}x + b_{1}y + c_{1} = 0

a_{2}x+ b_{2}y + c_{2} = 0

Following are the cases:

(i) If \(\frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}}\) then the lines are intersecting lines and intersect at one point. In this case, the pair ofÂ linear equations in consistent.

(ii) If \(\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}\), then the lines are coincident. In this case, the pair of linear equation is consistentÂ (dependent)

(iii) If \(\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}}\) then the lines are parallel to each other. In this case, the pair of linear equations inconsistent.