# NCERT Solutions for Class 10 Maths Chapter 3 Pair of Linear Equations in Two Variables

(Last Updated On: April 9, 2023)

NCERT Solutions for Class 10 Maths Chapter 3 Pair of Linear Equations in Two Variables Introduction : A pair of linear equations in two variables is a set of two equations in the form:

ax + by = c dx + ey = f

where x and y are variables .

Where x and y stand for the coordinates of a point on the line, each equation represents a line in a two-dimensional coordinate plane.

The set of x and y values that simultaneously satisfy both equations are the solution to the pair of linear equations in two variables. The intersection of the two lines in the coordinate plane is represented geometrically by the solution’s point or points.

There are three possible types of solutions for a pair of linear equations in two variables:

• Unique solution: The two lines intersect at one point, and there is only one solution to the system of equations.
• No solution: The two lines are parallel and do not intersect, so there is no solution to the system of equations.
• Infinite solutions: The two lines are coincident, meaning they overlap each other, so there are infinitely many solutions to the system of equations.

## NCERT Solutions for Class 10 Maths Chapter 3 Pair of Linear Equations in Two Variables

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Pair of Linear Equations Class 10 has total seven exercises consisting of 55 problems. Problems will be based on concepts like linear equations in two variables, algebraic methods for solving linear equations, elimination method, cross-multiplication method time and work, age, boat current and reducible equations for a pair of linear equations, These answers will give you ease in solving problems related to linear equations.

### Topics and Sub Topics in Class 10 Maths Chapter 3 Pair of Linear Equations in Two Variables:

 Section Name Topic Name 3 Pair of Linear Equations in Two Variables 3.1 Introduction 3.2 Pair Of Linear Equations In Two Variables 3.3 Graphical Method Of Solution Of A Pair Of Linear Equations 3.4 Algebraic Methods Of Solving A Pair Of Linear Equations 3.4.1 Substitution Method 3.4.2 Elimination Method 3.4.3 Cross-Multiplication Method 3.5 Equations Reducible To A Pair Of Linear Equations In Two Variables 3.6 Summary

### What is Pair of Linear Equations in Two Variables

A pair of linear equations in two variables is a group of two equations, each of which is a linear equation, commonly denoted as x and y.

following are set of two linear equations:

a1x + b1y = c1 a2x + b2y = c2

The solution of the system of equations is the values of x and y that satisfy both equations simultaneously.

Geometrically, a pair of linear equations in two variables represents two straight lines in a two-dimensional plane. The solution to the system of equations is the point(s) of intersection of these two lines.

There are three possible types of solutions for a pair of linear equations in two variables:

• Unique solution: The two lines intersect at one point, and there is only one solution to the system of equations.
• No solution: The two lines are parallel and do not intersect, so there is no solution to the system of equations.
• Infinite solutions: The two lines are coincident, meaning they overlap each other, so there are infinitely many solutions to the system of equations.

Pair of linear equations in two variables are used in various fields, including mathematics, physics, engineering, economics, and business. They are essential in solving optimization problems, making predictions, and modeling real-world scenarios.

### NCERT Solutions for Class 10 Maths Chapter 3 Pair of Linear Equations in Two Variables

Pair of Linear Equations Class 10 has total seven exercises consisting of 55 problems. Problems will be based on concepts like linear equations in two variables, algebraic methods for solving linear equations, elimination method, cross-multiplication method time and work, age, boat current and reducible equations for a pair of linear equations, These answers will give you ease in solving problems related to linear equations.

Topics and Sub Topics in Class 10 Maths Chapter 3 Pair of Linear Equations in Two Variables:

#### NCERT Solutions for Class 10 Maths Chapter 3 Pair of Linear Equations in Two Variables Ex 3.1

NCERT Solutions for Class 10 Maths Chapter 3 Pair of Linear Equations in Two Variables Ex 3.1 is part of the NCERT Solutions for Class 10 Maths. Here we have given NCERT Solutions for Class 10 Maths Chapter 3 Pair of Linear Equations in Two Variables Exercise 3.1

 Board CBSE Textbook NCERT Class Class 10 Subject Maths Chapter Chapter 3 Chapter Name Pair of Linear Equations in Two Variables Exercise Ex 3.1 Number of Questions Solved 3 Category NCERT Solutions

### Ex 3.1 Class 10 Maths Question 1.

Aftab tells his daughter, “Seven years ago, I was seven times as old as you were then. Also, three years from now, I shall be three times as old as you will be”. Isn’t this interesting? Represent this situation algebraically and graphically.

Solution: Let the current age of Aftab’s daughter be x years.

According to the problem, “Seven years ago, I was seven times as old as you were then”. So, Aftab’s age 7 years ago was (x+7), and his age was 7 times his daughter’s age at that time, which was (x-7). This can be represented algebraically as:

Aftab’s age 7 years ago = x + 7 Aftab’s age 7 years ago = 7(x-7)

Simplifying the above equations, we get: x + 7 = 7x – 49 6x = 56 x = 9.33 (approx.)

Therefore, Aftab’s daughter’s current age is approximately 9.33 years.

In three years, I’ll be thrice your age, the speaker declared. Aftab will therefore be (x+3) years old and his daughter will be (x+3) years old in three years. This is represented as follows in algebra:

Age of Aftab in three years equals x + 3 Age of Aftab in three years equals 3(x+3)

With the following equations simplified, we obtain: x + 3 = 3x + 9. 2x = -6 x = -3 (which is not possible)

Because the value of x is impossible, the issue statement appears to include a logical mistake.

Graphically, we can represent the two equations as two straight lines on a coordinate plane. Let x represent the daughter’s age, and y represent Aftab’s age.

The first equation, “Seven years ago, I was seven times as old as you were then”, can be rewritten as:

y – 7 = 7(x – 7)

Simplifying this equation, we get:

y = 7x – 42

The second equation, can be rewritten as:

y + 3 = 3(x + 3)

Simplifying this equation, we get:

y = 3x + 6

We can plot these two lines on a coordinate plane and find their point of intersection, which represents the solution to the system of equations. However, since the value of x is not possible in this case, we cannot graphically represent this situation. ### Ex 3.1 Class 10 Maths Question 2.

The coach of a cricket team buys 3 bats and 6 balls for ₹ 3900. Later, she buys another bat and 3 more balls of the same kind for ₹1300. Represent this situation algebraically and geometrically.

Solution: Let cost of one bat is x and cost of one ball is y.

According to the problem, the coach buys 3 bats and 6 balls for ₹ 3900. This can be represented algebraically as:

3x + 6y = 3900 —(1)

Later, she buys another bat and 3 more balls of the same kind for ₹1300. This can be represented algebraically as:

x + 3y = 1300 —(2)

We can solve this system of equations using any method of solving linear equations. For instance, we can solve equation (2) for x and substitute that expression into equation (1), then solve for y. Alternatively, we can solve equation (1) for x and substitute that expression into equation (2), then solve for y.

Solving equation (2) for x, we get:

x = 1300 – 3y

Substituting this expression into equation (1), we get:

3(1300 – 3y) + 6y = 3900

Simplifying the above equation, we get:

3900 – 9y + 6y = 3900

-3y = 0

y = 0

Substituting this value of y into equation (2), we get:

x + 3(0) = 1300

x = 1300

As a result, one bat costs RS1300, .

The cost of one ball cannot be zero, hence this conclusion seems strange. The problem statement can contain a typo or other error.

The equations No. (1) and (2) are straight lines on a coordinate plane. Let x stand in for the price of one bat, and y for the price of one ball.

The first equation, 3x + 6y = 3900, can be rewritten in slope-intercept form as:

y = (-1/2)x + 650

The second equation, x + 3y = 1300, can be rewritten in slope-intercept form as:

y = (-1/3)x + 433.33

We can plot these two lines on a coordinate plane and find their point of intersection, which represents the solution to the system of equations

Draw a graph for above question for x= 100,400,700

We can use the equations we previously derived to discover the values of y that match to the values of x given in the question:

Y = (3900-3x)/6 = (3900-300)/6 = 600/6 = 100 for x = 100.

Y = (3900-3x)/6 = (3900-1200)/6 = 2700/6 = 450 for x = 400.

Y = (3900-3x)/6 = (3900-2100)/6 = 1800/6 = 300 for x = 700.

we can map them on a coordinate plane:

The points (100, 100), (400, 450), and (700, 300) represent the solutions to the system of equations for the specific values of x given in the question. ### Ex 3.1 Class 10 Maths Question 3.

The cost of 2 kg of apples and 1 kg of grapes on a day was found to be ₹160.  After a month, the cost of 4 kg of apples and 2 kg of grapes is ₹300. Represent the situation algebraically and geometrically.

Solution: et the cost of one kg of apples be ‘a’ and the cost of one kg of grapes be ‘g’.

According to the problem, “the cost of 2 kg of apples and 1 kg of grapes on a day was found to be ₹160”, so we have the equation:

2a + 1g = 160

After a month, “the cost of 4 kg of apples and 2 kg of grapes is ₹300”, so we have the equation:

4a + 2g = 300

These two equations can be solved , values of “a” and “g,” :

2a + 1g = 160 (equation 1) 4a + 2g = 300 (equation 2)

Multiplying equation 1 by 2 and subtracting it from equation 2, we get:

2a = 80

Solving for ‘a’, we get:

a = 40

Substituting the value of ‘a’ in equation 1, we get:

2(40) + g = 160

Simplifying this equation, we get:

g = 80

Therefore, the cost of 1 kg of apples 40 RS , and cost of 1 kg grapes is RS 80.

We can represent this situation geometrically by graphing the two equations on a coordinate plane.

The first equation, “the cost of 2 kg of apples and 1 kg of grapes on a day was found to be ₹160”, can be rewritten as:

g = 160 – 2a

The second equation, “the cost of 4 kg of apples and 2 kg of grapes is ₹300”, can be rewritten as:

g = 150 – 2a

We can plot these two lines on a coordinate plane and find their point of intersection, which represents the solution to the system of equations.  System of a Pair of Linear Equations in Two Variables
A system of a pair of linear equations in two variables is a set of two equations with two variables, where each equation is linear in nature.

a₁x + b₁y = c₁

a₂x + b₂y = c₂

where x and y are the variables.

The goal is to find a unique solution that satisfies both equations simultaneously.

There are three possibilities for the solution of such a system:

Unique solution: The two lines intersect at exactly one point, which is the solution of the system.
No solution: The two lines are parallel and do not intersect. In this case, there is no solution for both equations.
Infinitely many solutions: The two lines coincide with each other, meaning that they are the same line. In this case, there are infinitely many solutions that satisfy both equations.

Solving a system of a pair of linear equations in two variables involves finding values of x and y that satisfy both equations. There are various methods to solve such systems, including substitution method, elimination method, and graphing method.

Representation of Linear Equation In Two Variables

A linear equation in two variables is an equation that can be written in the form:

ax + by = c

The degree of both x and y is 1, meaning that each variable is raised to the power of 1.

A linear equation with two variables may be visualised in a variety of ways, including:

On a two-dimensional coordinate plane, a straight line can be used to visually depict a two-variable linear equation. Marking two points on the line and connecting them with a straight line allows us to graph a linear equation. There is a solution to the issue somewhere along this line.
A linear equation looks like this in slope-intercept form:.

y = mx + b

where b and m stand in for the line’s y-intercept and y-slope, respectively.

Point-slope form: In point-slope form, a linear equation looks like:

y – y₁ = m(x – x₁)

where (x1, y1) is a point on the line and m is an angle of the line’s slope.

Standard form: The following is an example of a linear equation in standard form:

axe + by = c

A linear equation in two variables can be represented using any of these formats, and depending on the circumstance, each format has pros and cons.

Graphing a Linear Equation with Two Variables

The steps below must be taken in order to graph a linear equation involving two variables:

If at all possible, write the equation in slope-intercept form rather than standard form.
Find the line’s y-intercept, which is where it crosses the y-axis. The slope-intercept form’s value of b or the standard form’s ratio of c/b serve as the y-intercept indicators.
Locate a different line point. Utilizing the slope, which is determined by the coefficient, is one method for doing this.

Here’s an example:

Plot the line y = 2x + 3 on the graph.

This equation is already in slope-intercept form, where the slope is 2 and the y-intercept is 3.
The y-intercept is (0, 3).
To find another point, we can start from the y-intercept and move up by 2 units (since the slope is 2) and right by 1 unit (since the change in x is 1). This gives us the point (1, 5).
Plot the two points (0, 3) and (1, 5) on the graph, then draw a straight line connecting them.

The graph that results should resemble a diagonal line that traverses the points (0, 3) and (1, 5).

Note that if the equation is not in slope-intercept form, we can still plot it by rearranging it into the standard form or by solving for y and then following the above steps.

Method of Solution of a Pair of Linear Equations in Two Variables

There are different methods to solve a pair of linear equations in two variables, including:

Graphical Method: This method involves plotting the two equations on the same graph and finding the point of intersection, which is the solution to the system. This method is simple but may not be accurate when the solution involves decimal or fractional values.
Substitution Method: This method involves solving one of the equations for one variable and substituting that expression into the other equation.

Elimination Method: In this approach, the two equations are added or subtracted in a way that gets rid of one of the variables, creating a single equation that can be solved.

Matrix Method: This method involves writing the coefficients of the variables as a matrix and performing row operations to convert the matrix to reduced row echelon form. As a result, a straightforward system of equations results, and the matrix can be used to read the answer.

A pair of linear equations in two variables can be solved using any of these approaches, however depending on the situation, some may be more effective or appropriate.

Consistency and Nature of the Graphs

The consistency and nature of the graphs of a pair of linear equations in two variables depend on whether the equations have a common solution, no solution, or infinitely many solutions.

Consistent System: A pair of linear equations is said to be consistent if they have a common solution. As a result, a straightforward system of equations results, and the matrix can be used to read the answer.

A pair of linear equations in two variables can be solved using any of these approaches, however depending on the situation, some may be more effective or appropriate.

Dependent System: A pair of linear equations is said to be dependent if they have infinitely many solutions. Graphically, this means that the two lines coincide and overlap each other. In this case, any point on the line is a solution to the system.

The nature of the graphs of a pair of linear equations can also provide information about the equations themselves.

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### Where we use Pair of Linear Equations in Two Variables in daily life

Pair of Linear Equations in Two Variables are used in various aspects of our daily life, such as:
1. Budgeting: While planning a budget, we often have to solve linear equations to determine the cost of various expenses and how much we can afford to spend.
2. Business: Linear equations can be used in business to calculate the profit or loss of a company, determine the break-even point, and analyze trends in sales and revenue.
3. Engineering: Linear equations are used in engineering to model physical systems and design structures such as bridges and buildings.
4. Transportation: Linear equations are used to determine the distance, time, and speed required to travel from one place to another, which is essential in transportation planning.
5. Sports: Linear equations can be used in sports to analyze player performance, determine the winning strategy, and predict future outcomes.
6. Science: Linear equations are used in scientific research to model and analyze various phenomena such as the growth of populations, the spread of diseases, and the rate of chemical reactions.
7. Agriculture: Linear equations can be used in agriculture to determine the optimal amount of fertilizer or water needed to grow crops, and to predict crop yields.

Pair of Linear Equations in Two Variables is a fundamental concept that has applications in many different fields and aspects of our daily lives.

## FAQs About Pair of Linear Equations in Two Variables

Here are some frequently asked questions about Pair of Linear Equations in Two Variables:

1. What is a Pair of Linear Equations in Two Variables? A Pair of Linear Equations in Two Variables is a system of two equations that contain two variables. The general form of a pair of linear equations in two variables is ax + by = c and dx + ey = f.
2. What is the importance of solving Pair of Linear Equations in Two Variables? Pair of Linear Equations in Two Variables is important in solving real-life problems related to business, science, engineering, economics, and many other fields. By solving these equations, we can determine the values of the variables and make predictions or decisions based on the solutions.
3. What are the methods of solving Pair of Linear Equations in Two Variables? There are several methods to solve Pair of Linear Equations in Two Variables, including substitution method, elimination method, graphical method, and matrix method.
4. Can Pair of Linear Equations in Two Variables have more than one solution? Yes, a Pair of Linear Equations in Two Variables can have one unique solution, infinite solutions, or no solution at all. The number of solutions depends on the coefficients of the equations and their relationship to each other.
5. How do Pair of Linear Equations in Two Variables relate to straight lines? A Pair of Linear Equations in Two Variables represents two straight lines in a plane. The solution of the system represents the point where the two lines intersect. If the lines are parallel, the system has no solution, and if they are the same line, the system has an infinite number of solutions.
6. What are the applications of Pair of Linear Equations in Two Variables in real life? Pair of Linear Equations in Two Variables has numerous applications in real-life situations, including budgeting, business, engineering, transportation, sports, science, and agriculture. By using these equations, we can model and analyze various phenomena and make informed decisions based on the solutions