Pair of Linear Equations in two Variables Class 10 ||Maths|| Chapter 3 NCERT Notes

1. Linear Equation in Two Variables

A linear equation in two variables $x$ and $y$ can be written in the form:

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

Where:

$a_{1}$ , $b_{1}$ , and $c_{1}$ are real numbers
$x$ and $y$ are variables

For example: $2 x + 3 y = 5$ is a linear equation in two variables.

2. Solution of a Pair of Linear Equations

A pair of linear equations in two variables can be written as:

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

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

The solutions of the pair of equations are the values of $x$ and $y$ that satisfy both equations simultaneously.

3. Graphical Method of Solution

The graphical method involves plotting both equations on the graph. The nature of the solution depends on how the lines representing the equations intersect.

Intersecting Lines:
- If the two lines intersect at a point, the system has a unique solution.
- The point of intersection gives the solution $(x, y)$ .
Parallel Lines:
- If the lines are parallel, there is no solution.
- This is called an inconsistent system.
Coincident Lines:
- If the lines coincide (overlap completely), there are infinitely many solutions.
- This is called a dependent system.

4. Algebraic Methods of Solving Pair of Linear Equations

There are three algebraic methods to solve a pair of linear equations:

a. Substitution Method

Steps:

Solve one equation for one variable in terms of the other.
Substitute this expression in the second equation.
Solve the resulting equation for one variable.
Substitute the value into any original equation to find the value of the other variable.

b. Elimination Method

Steps:

Multiply the equations (if necessary) to make the coefficients of one variable the same.
Add or subtract the equations to eliminate one variable.
Solve the resulting equation for the remaining variable.
Substitute this value into one of the original equations to find the other variable.

c. Cross Multiplication Method

For equations of the form:

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

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

The solution can be found using:

\frac{x}{b_{1} c_{2} - b_{2} c_{1}} = \frac{y}{c_{1} a_{2} - c_{2} a_{1}} = \frac{1}{a_{1} b_{2} - a_{2} b_{1}}

This method directly gives the values of $x$ and $y$ .

5. Conditions for Consistency

To determine whether a pair of linear equations has a solution, consider the ratios of the coefficients:

$\frac{a_{1}}{a_{2}}$ , $\frac{b_{1}}{b_{2}}$ , and $\frac{c_{1}}{c_{2}}$

Unique Solution (Consistent System): If $\frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}}$ , the system has a unique solution (intersecting lines).
No Solution (Inconsistent System): If $\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}}$ , the system has no solution (parallel lines).
Infinitely Many Solutions (Dependent System): If $\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}}$ , the system has infinitely many solutions (coincident lines).

6. Applications of Linear Equations

Linear equations in two variables can be used to solve real-life problems involving:

Numbers, ages, costs
Situations involving relationships between quantities
Motion and geometry problems

Steps to Solve Word Problems:

Convert the problem into a pair of linear equations.
Solve using the graphical or algebraic methods.
Interpret the solution in the context of the problem.

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