Pair of Linear Equations in Two Variables

Linear Equation in two variables are equations with only two variables and the exponent of the variable is 1. This system of equations can have a unique solution, no solution, or an infinite solution according to the given initial condition. Linear equations are used to describe a relationship between two variables. Sometimes in some situations, we don’t know the values of the variables we want to observe. So, then we formulate the equations describing how they behave and solve them. The Number of equations obtained should be equal to the number of variables.

Let’s learn about the Pair Of Linear Equation In Two Variables and their solution in this article.

Pair of Linear Equations In Two Variables

A linear equation is defined as:

ax + by + c = 0

where,

• a, b and c are real numbers
• a and b both are not zero

A pair is formed by two such linear equations. It can be represented as:

a₁x + b₁y + c₁ = 0

a₂x + b₂y + c₂ = 0

where,
a₁, b₁, c₁, a₂, b₂ and c₂ are real numbers. 

Since a linear equation represents a line on the Cartesian plane. A pair represents two lines on the Cartesian plane. The solution to this system will be the points that satisfy both of these equations. There are three possibilities in such a system.

What are Pair of Linear Equations in Two Variables?

Pair of linear equations in two variables refers to a system of two equations with two variables, typically x and y. These equations are linear, meaning that each variable is raised to the power of one and doesn’t involve any other operations like multiplication or division. The general form of a pair of linear equations in two variables is:

General Form: ax + by = c​

where a, b, and c are constants, and x and y are the variables. The objective when dealing with such a pair of equations is to find the values of x and y that satisfy both equations simultaneously.

This often involves techniques like substitution, elimination, or graphical methods. Pair of linear equations in two variables are extensively used in various fields including mathematics, physics, economics, and engineering to model and solve real-world problems involving two unknown quantities.

No Solution

If lines are parallel to each other then they have no solution.

For example, 2x + 3y = 11 and 2x + 3y = 13 are parallel lines and they have no solution.

Unique Solution

If two lines intersect each other at any point then they have a unique solution.

For example, 2x + 3y = 11 and 3x + 2y = 11 are intersecting lines and they have a unique solution.

Infinitely Many Solutions

If two lines are overlapping with each other then they have infinite solutions.

For example, 2x + 3y = 11 and 4x + 6y = 22 are overlapping lines and they have infinite solutions.

Solution of Linear Equation in Two Variables

Pair of Linear Equations in Two Variables Formulas

The pair of linear equations in two variables can be represented as:

1. Standard Form: ax+by=c
2. Slope-Intercept Form: y=mx+b
3. Point-Slope Form: y−y₁​=m(x−x_1​)

Where:

• a, b, and c are constants, and a and b are not both zero.
• m represents the slope of the line.
• b is the y-intercept (the point where the line intersects the y-axis).
• x₁​ and y₁​ are coordinates of a point on the line.

Standard Form

The standard form of a linear equation in two variables, x and y, is represented as ax+by=c, where a, b, and c are constants, and a and b are not both zero. In this form, a and b represent the coefficients of the x and y terms, respectively, and c is the constant term. Standard form is beneficial for solving equations using methods like elimination and substitution, especially when working with systems of equations.

Slope-Intercept Form

The slope-intercept form of a linear equation is written as y=mx+b, where m represents the slope of the line, and b is the y-intercept, the point where the line intersects the y-axis. In this form, m determines the inclination or steepness of the line, while b represents the y-coordinate of the point where the line crosses the y-axis. Slope-intercept form is particularly convenient for graphing linear equations, as it directly provides information about the slope and y-intercept.

Point-Slope Form

The point-slope form of a linear equation is expressed as y−y₁​=m(x−x₁​), where m is the slope of the line, and (x₁​,y₁​) is a point on the line. This form is derived from the slope formula, (y₂​−y₁​)/(x₂​−x₁​), where m represents the rate of change between two points on the line. Point-slope form is useful for finding the equation of a line given its slope and a point on the line. It offers flexibility in finding the equation of a line passing through a specific point with a given slope.

Representation of Pair of Linear Equation in Two Variables

The pair of linear equations can be solved and represented by two methods which include,

• Graphical Method
• Algebraic Method

The general form of a pair of linear equations in two variables is:

a₁x + b₁y + c₁ = 0

a₂x + b₂y + c₂ = 0

where,
a₁, b₁, c₁, a₂, b₂ and c₂ are real numbers
a₁² + b₁² ≠ 0
a₂²+ b₂² ≠ 0

The pair of linear equations have three conditions,

• If a₁/a₂ ≠ b₁/b₂ the pair of linear equations is consistent.
• If a₁/a₂ = b₁/b₂ ≠ c₁/c₂ the pair of linear equations is inconsistent.
• If a₁/a₂ = b₁/b₂ = c₁/c₂ the pair of linear equations is dependent and consistent.

If we compare the coefficients a₁, b₁, c₁, a₂, b₂ and c₂  of different pairs of equations which have a unique solution, infinite solutions, and no solutions. We obtain the results given below in the table. 

For two lines:

a₁x + b₁y + c₁ = 0

a₂x + b₂y + c₂ = 0

Graphical Representation	Algebraic Interpretation	Conditions
One intersection	Unique Solution	a1/a2 ≠ b1/b2
Coincident Lines	Infinitely Many Solutions	a1/a2 = b1/b2 = c1/c2
No intersection/ Parallel Lines	No Solution	a1/a2 = b1/b2 ≠ c1/c2

Graphical Representation

In this method, we represent the equations on the graph and find out their intersection through it. We look for the points which are common to both lines, sometimes there is only a point like that, but it might also happen that there are no solutions or infinite solutions. 

Example: Find out the intersection of the following lines. 

3x + 5y = 6 

x + y = 2

Solution: 

We will plot both the lines on the graph. 

Graphical Representation of linear equation

These lines intersect at (2,0). 

Algebraic Methods of Solving a Pair of Linear Equations

Various methods of Solving a Pair of Linear Equations are,

• Substitution Method
• Elimination Method
• Cross-Multiplication Method 

Substitution Method

In this method, we use one equation to express a variable in terms of the other variable thereby reducing the number of variables in the equation. Then we substitute that expression into the other equation that is given to us.

Example: Solve the following pair of equations with the substitution method. 

x + y = 3 

3x + y = 16

Solution: 

Let’s pick the equation 

x + y = 3 

x = 3 – y 

Substituting the value of x in the other equation,

3x + y = 16 

3(3 – y) + y = 16 

9 – 3y + y = 16 

-2y =7 

y = -7/2 

Elimination Method 

This method is sometimes more convenient than the substitution method. In this method, we eliminate one variable by multiplying and adding equations with suitable constants, this is done to eliminate one variable, and when the equation is left with just one variable, it can easily be solved.

Example: Solve the following equations with the elimination method. 

x + y = 3 

x – y = 5

Solution:

We have two equations, 

x + y = 3  …….(1)

x – y = 5 …… (2)

Adding the equation (1) and (2) to eliminate the variable -y. 

2x = 8 

x = 4 

Substituting the value of x in equation (1) 

4 + y = 3 

y = -1

Cross-Multiplication Method 

This method looks more complex than the other methods, but it is one of the most efficient ways to solve linear equations. Let’s say the two lines whose equation is:

a₁x + b₁y + c₁ = 0

a₂x + b₂y + c₂ = 0

In this cross-multiplication method,

Cross-Multiplication Method

The solution is given by:

x/(b₁c₂-b₂c₁) = y/(c₁a₂-c₂a₁) = 1/(a₁b₂-a₂b₁)

Example: Solve the following equations with the cross-multiplication method. 

2x + 3y = 46 

3x + 5y = 74 

Solution:

a₁ = 2, a₂ = 3, b₁ = 3, b₂ = 5, c₁ = 46 and c₂ = 74

x/(b₁c₂-b₂c₁) = y/(c₁a₂-c₂a₁) = 1/(a₁b₂-a₂b₁)

x/(3-(-74)-5(-46)) = y/(-46(3)-(-74)2) = 1/(2(5)-3(3))

x/8 = y/10 = 1/1

x = 8 and y = 10

Solved Examples on Linear Equations in Two Variables

Example 1: Solve the following pair of equations graphically: 

2x + 3y = 46 

3x + 5y = 74 

Solution: 

We need to plot them of graph separately and then look at their intersection. 

Graph Plotting of Example 1

This graph intersects are (10,8).

Example 2: Solve the following pair of linear equations with the substitution method. 

5x + 4y = 20 

x + 2y = 4

Solution: 

We have to solve these two equations 

5x + 4y = 20 

x + 2y = 4

Let’s we pick the second equation, 

x = 4 – 2y

Now substituting the value of x in the other equation. 

5(4 – 2y) + 4y = 20 

20 – 10y + 4y = 20 

-6y = 0 

y = 0 

Finding out the value of x by substituting the value of y in the equation, 

x = 4 – 2y 

x = 4 

(4, 0) is the solution to this pair of linear equations. 

Example 3: Solve the following equations with the elimination method. 

4x + 5y = 20 

8x + 2y = 5

Solution: 

Let the equations be, 

4x + 5y = 20 ……..(1)

8x + 2y = 5 ….. (2)

We need to eliminate one of the variables here from these two equations, 

Multiply equation (1) with 2 and subtract it with (2). 

2 x(1) -(2) 

8x + 10y = 40 ….. 2 x(1)

8x + 2y = 5 …..(2) 

Subtracting both of these, 

8y = 35 

y = 35/8

Substituting this value in equation (1) 

4x+5(35/8​)=20 = 4x+175/8=20

4x = 20 –175/8

4x = −15​/8

x = −15​/32

Pair of Linear Equations in Two Variables Class 10 Extra Questions

1. Solve the pair of equations: x−2y=5 and 2x+y=4.
2. Find the solution to the system of equations: 4x+3y=10 and 2x−y=3.
3. Determine whether the equations 5x−2y=8 and 10x−4y=16 have a unique solution, infinitely many solutions, or no solution.
4. Solve the system of equations: 3x+2y=7 and 2x−3y=1.
5. Find the values of x and y that satisfy the equations: 2x+5y=17 and 3x−y=5.

Pair of Linear Equations in Two Variables Solutions

Get free NCERT Solutions for Class 10 Maths Chapter 3 – Pair of Linear Equations in Two Variables and develop an easy approach to solving problems related to this chapter. In NCERT Class 10 Maths, Chapter 3 titled “Pair of Linear Equations in Two Variables” covers various themes including an introduction to linear equations, understanding pairs of linear equations, representing them graphically, converting equations into linear form, and applying these equations to solve practical problems in real-world scenarios.

FAQs on Linear Equations in Two Variables

What is linear equation in two variables?

A linear equation is an equation with degree 1. A linear equation in two variables is a type of linear equation in which there are 2 variables present. For example, 4x – 3y = 5, 3a – b = 4 etc.

How to solve linear equation in two variables?

Linear Equations in Two Variables are solved by two methods,

• Graphical Method
• Algebraic Method

What are examples of linear equations in two variables?

The standard form for linear equations in two variables is ax + by = c. For example, 4x + 7y = 5

How many solutions are there for linear equations in two variables?

A linear equation in two variables can have a unique solution, no solution and an infinitely solution.

What are the coefficients of equation 3x – 6y = -13?

The coefficient of x in the given equation is 3 and the coefficient of y in the equation is -6.

What is the constant of the equation 3x – 6y=-13?

The constant of the equation 3x – 6y = -13 is -13.

If the pair of linear equation in two variables are inconsistent then how many solution do they have?

If the pair of linear equations in two variables are inconsistent, they have no solution.