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

Pair of linear equation in two variables class 10 notes provided here are of extreme importance to the class 10 students as they can now learn the concepts easily and revise them quickly. These CBSE chapter 3 notes are given in a short and crisp way while covering all the important concepts included in this chapter. Even examples and practice questions are provided which will ensure the students to have in-depth knowledge about the pair of linear equations in two variables topic.

This notes for chapter 3 include the following points-

• Meaning of pair of linear equations in two variables
• Conditions for consistent, inconsistent, and dependent
• Methods of representing and solving linear equations in two variables
• Example questions
• Practice questions
• Articles related to pair of linear equations

## Definition of Pair of Linear Equations in Two Variables

Two linear equations are known as pair of two linear equations in two variables if they have two same variables in them. The general form of a pair of linear equations is given by-

$a_{1}x+b_{1}y+c_{1}=0$ $a_{2}x+b_{2}y+c_{2}=0$

Here, $a_{1},a_{2},b_{1},b_{2},c_{1},c_{2}$ are all real numbers and,

$a_{1}^{2}+b_{1}^{2}\neq 0,\, and\, a_{2}^{2}+b_{2}^{2}\neq 0,$

### Conditions for consistent, inconsistent, and dependent

For a pair of linear equations, suppose, $a_{1}x+b_{1}y+c_{1}=0\, and\, a_{2}x+b_{2}y+c_{2}=0$, three conditions can arise which determines the type of that pair of linear equations.

 Type Condition Consistent $\frac{a_{1}}{a_{2}}\neq \frac{b_{1}}{b_{2}}$ Inconsistent $\frac{a_{1}}{a_{2}}\neq \frac{b_{1}}{b_{2}}\neq \frac{c_{1}}{c_{2}}$ Dependent and Consistent $\frac{a_{1}}{a_{2}}= \frac{b_{1}}{b_{2}}= \frac{c_{1}}{c_{2}}$

Representation and Solving of Pair of Linear Equations in Two Variables

There are two methods for representing and solving linear equations in two variables which are-

1. Graphical Method
2. Algebraic Method

Graphical Method:

For the graphical method, any pair of linear equation is represented by two lines. From the intersection of those two lines, the type of equations and their solutions can be determined by the following way-

 Condition Type Solutions Lines intersecting at a point Consist The point of intersection Lines coinciding Dependent (consistent) Infinite solutions (each point is a solution) Lines are parallel Inconsistent No solution

Algebraic Method:

There are three methods of solving a pair of linear equation by algebraic method which are:

The details of each of these methods are explained in the linked article along with several solved examples for better understanding.

### Example:

Question:

The cost of 2 pencils and 3 erasers is Rs 9 and the cost of 4 pencils and 6 erasers is Rs 18. Find the cost of each
pencil and each eraser.

Solution :

The pair of linear equations formed were:

2x + 3y = 9 ___________________ (1)
4x + 6y = 18 __________________ (2)

We first express the value of x in terms of y from the equation 2x + 3y = 9, to get

$x=\frac{9-3y}{2}$

Now we substitute this value of x in Equation (2), to get

$\frac{4(9-3y)}{2}+6y=18$

i.e., 18 – 6y + 6y = 18
i.e., 18 = 18

This statement is true for all values of y.

### Practice Questions

1. Solve the following equations 2x + 3y = 11 and 2x – 4y = – 24. Also, find the value of ‘m’ for which y = mx + 3.
2. Mr. X and Mr. Y has the income ratio of 9 : 7 and the expenditure ratio of 4 : 3. If bothe Mr. X and mr. Y manages of Rs. 2000 per month individually, calculate their monthly incomes.
3. Calculate the values of “k” for which the pair of equations has unique solution?
4x + ky + 8 = 0
2x + 2y + 2 = 0

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