CBSE Class 10 Mathematics Chapter 3 - Pairs of linear equations in two variables - Notes

CBSE Class 10 Mathematics  Chapter 3 - Pairs of linear equations in two variables - Notes

What is pair of linear equations with two variables?

A pair of linear equations in two variables is a system of two equations that involve two variables and are both of first degree (i.e., they have an exponent of 1 for each variable). The general form of a pair of linear equations in two variables is:

a₁x + b₁y + c₁ = 0

a₂x + b₂y + c₂ = 0

Here, x and y are the variables, and a₁, b₁, c₁ and a₂, b₂, c₂ are coefficents.

The goal when dealing with such a system is to find the values of x and y that satisfy both equations simultaneously. This means you're looking for a point (x, y) that lies on both lines represented by these equations.

Graphical Method of Solution of a Pair of Linear Equations:

Inconsistent Pair of Equations:

• A pair of linear equations that has no solution is referred to as an inconsistent pair of linear equations. In this two lines are parallel and do not intersect
• When the lines are parallel, it means they have no common points of intersection, and thus there is no solution to the system. In this case,

Consistent Pair of Equations:

• A pair of linear equations in two variables that has a unique solution is known as a consistent pair of linear equations. In this a single point where the two lines intersect.
• When the lines represented by the equations a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 intersect, it means there is a unique solution (x, y) that satisfies both equations. In this case, it's not necessary that

• as the lines can still intersect even if 

The key is that the lines do intersect at a single point.

Dependent Pair of Equations:

• A pair of linear equations in two variables that are equivalent and share infinitely many distinct common solutions is termed a dependent pair of linear equations. Importantly, a dependent pair of equations is always consistent. In this the two lines are coincident, meaning they lie on top of each other.
• When the lines are coincident, it means they are the same line and have infinitely many common solutions. In this case,

Method for Solution:

• Write the Equations in Standard Form like : Ax + By = C, where A, B, and C are constants.
• Isolate 'y' in Each Equation: Rearrange both equations to solve for 'y' in terms of 'x'-

Equation 1: y = (C - Ax) / B

Equation 2: y = (F - Dx) / E

Create a Table of Values:

• Create a table with two columns, one for 'x' and one for 'y'.
• Choose a range of 'x' values and calculate the corresponding 'y' values using the equations you obtained in step 2.

Plot the Lines:

• Plot the points (x, y) as per table of values for both equations. Connect the points with straight lines. These lines represent the two equations.

Identify the Intersection Point:

• The solution to the system of equations is the point where the two lines intersect. This is the (x, y) value that satisfies both equations.

Read the Coordinates:

• Read the coordinates of the intersection point. These coordinates are the values of 'x' and 'y' that solve the system of equations.

Check for Coincident Lines or No Solution:

• If the lines are coincident (they overlap), it means there are infinitely many solutions.
• If the lines are parallel and never intersect, it means there is no solution to the system.

Write the Solution:

• Write the solution as (x, y), where 'x' and 'y' are the coordinates of the intersection point you found.

Example 1: x — 2y = 0   ;    3x + 4y = 20

Solution: For x — 2y = 0 table of value

x	—2	—1	0	1	2	3
y	—1	—0.5	0	0.5	1	1.5

For 3x + 4y = 20 table of value

x	—2	—1	0	1	2	3
y	6.5	5.75	5	4.25	3.5	2.75

By Plotting the Lines on graph:

As shown in graph the intersection point of the two equations is x = 4 and y = 2. So coordinates are (4,2) and both equation are consistent pair of linear equations which have a unique solution.

Example 2: To determine the nature of the solution for the given pair of equations through graphical analysis, we can ascertain whether they have no solution, a unique solution, or infinitely many solutions.

Solution: First of all we write these equations in standard form:

As given equation (i) is already in standard form.

For convert equation (ii) in standard form multiply this by 5

    Table of value for equation (i)

x	-3	-2	-1	0	1	2	3
y	-1.75	-1.125	-0.5	0.125	0.75	1.375	2

Table of value for equation (ii)

x	-3	-2	-1	0	1	2	3
y	-1.75	-1.125	-0.5	0.125	0.75	1.375	2

As we see, both equations are same and lines represented by them are coincident. So they have infinity many solutions.

Example 3: Champa went to a ‘Sale’ to purchase some pants and skirts. When her friends asked her how many of each she had bought, she answered, “The number of skirts is two less than twice the number of pants purchased. Also, the number of skirts is four less than four times the number of pants purchased”. Help her friends to find how many pants and skirts Champa bought.

Solution: Assume that:

• P = the number of pants Champa purchased
• S = be the number of skirts Champa purchased

First of all, we write these in standard equations form as per questions

S = 2P — 2 …………… (i)

S = 4x — 4 …………. (ii)

Make a table of value of above equation. For equation (i) table of value:

x	—2	—1	0	1	2	3
y	—6	—4	—2	0	2	4

For equation (ii) table of value:

x	—2	—1	0	1	2	3
y	—12	—8	—4	0	4	8

As shown in graph two line intersect each other on point (1,0). Its mean Champa purchased one Pent and zero shirt.