NCERT Solutions Class 10 Maths Chapter 3 Exercise 3.1

NCERT Solutions Class 10 Maths Chapter 3 Exercise 3.1 is important in understanding the concept of linear equations in two variables. This is an important topic for solving real-world problems through mathematical models. Through this exercise, students will understand how to write and solve linear equations, which is essential in understanding higher mathematics. With this knowledge, students of class 10 can more effectively solve problems in everyday life and see how algebra can be used to represent real-life situations well. So, let’s explore this exercise in detail. 

1.0Download NCERT Solutions for Class 10 Maths Chapter 3  Exercise 3.1 : Free PDF

2.0Introduction to Linear Equations in Two Variables

A linear equation in two variables is an algebraic equation which represents real-world problems by modelling relationships between two unknown quantities (usually represented with x and y). The degree or the highest power of these variables is 1. The equation has a general or standard form, written as: 

ax + by + c = 0

Here, a, b, and c are the real numbers where a and b0. These equations, when represented on the graph, form a straight line. The solution to a linear equation is a pair of values (x, y), which make the equation stand true. 

3.0Exercise Overview: Key Concepts 

Algebraic Representation of Linear Equations in Two Variables 

The most important part of linear equations in two variables is the formation of algebraic equations of real-life problems. These equations eventually help in the representation of these equations on the graph. To form an algebraic equation for linear equations in two variables, follow these steps:

Step 1: Identify the Variables

Step one is to identify the two values you are dealing with. These values will be x and y in most problems. For instance, if you are dealing with the number of apples (x) and the number of oranges (y), your variables will be x and y.

Step 2: Write and Translate the Relationship Between the Variables

In all real-world issues, there is a connection between the two variables. The relationship is usually termed in terms such as "total cost," "distance," or "time." For example, suppose the cost of purchasing x Giant Wheel rides, each costing Rs 3, and y games of Hoopla, costing Rs 4 each, which cost in total Rs 20, you can write this equation as: 

3x + 4y = 20

Step 3: Ensure the Correct Form

Ensure that the equation follows the standard form of linear equation in two variables, which is ax + by + c = 0

So, the above equation may be written as: 

3x + 4y - 20 = 0

Graphical Representation of Linear Equations in Two Variables 

Graphical Representation of a given linear equation in two variables results in a straight line on the coordinate plane. This can be done by following these steps while in the standard form: 

Step 1: Find Two or More Points

• To graph the equation on the graph, you must have at least two points that meet the equation.
• This is done by assigning values to x (or y) and determining the corresponding y (or x).
• Generally, for these values, we first put x = 0, determine the value of y and then put y = 0 and find the value of x.
• These coordinates (x, y) will provide you with points to plot on the graph.

Step 2: Plot the Points

After having two points (x, y), graph these points on a coordinate plane using the horizontal X-axis and vertical Y-axis).

Step 3: Draw the Line

After marking the points, draw a straight line through them. This is the linear equation.

Step 4: Label the Graph

Lastly, title your axes and indicate the equation on the graph.

Pair of Linear Equations in Two Variables

A system of two linear equations in two variables includes two linear equations with two variables, often x and y. The two equations are typically of the form: 

$a_{1}x+b_{1}y+c_1=0$

$a_{2}x+b_{2}y+c_2=0$

Here, a₁, b₁, and c₁ are the constants of the first linear equation, while a₂, b₂, and c₂ are the constants of the second equation, where a_1, a_2, b_1, \text{ and } b_2 \neq 0. This pair of equations is represented algebraically and graphically the same as the single linear equation. When representing graphically, a pair of linear equations can form three types of relationships: 

• The two lines will intersect at a single point.
• The two lines will not intersect at all, i.e., they are parallel.
• The two lines will be coincident (same lines)

4.0NCERT Class 10 Maths Chapter 3 Exercise 3.1: Detailed Solutions

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." Represent this situation algebraically and graphically.

Solution:

Let the present age of Aftab's daughter be x years.

Let the present age of Aftab be y years (y > x).

Seven years ago:

Aftab's age: y - 7

Daughter's age: x - 7

According to the problem: y - 7 = 7(x - 7)

Simplifying: y - 7 = 7x - 49

Equation 1: 7x - y - 42 = 0

Three years later:

 Aftab's age: y + 3

 Daughter's age: x + 3

 According to the problem: y + 3 = 3(x + 3)

 Simplifying: y + 3 = 3x + 9

 Equation 2: 3x - y + 6 = 0

 Therefore, the algebraic representation is:

  7x - y - 42 = 0

  3x - y + 6 = 0

When plotted on a graph, the intersection point of the two lines is (12, 42).

Thus, the present age of Aftab's daughter is 12 years, and the present age of Aftab is 42 years.

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:

Algebraic Representation:

Let the cost of 1 bat be ₹ x.

Let the cost of 1 ball be ₹ y.

According to the problem:

3x + 6y = 3900

x + 3y = 1300

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:

Let the cost of 1 kg of apples be ₹ x.

Let the cost of 1 kg of grapes be ₹ y.

According to the problem:

2x + y = 160

4x + 2y = 300

5.0Benefits of Studying NCERT Solutions Class 10 Maths Chapter 3 Exercise 3.1

• By practicing NCERT solutions, students can improve their calculation speed and minimize errors, which is crucial for scoring well in exams.
• Students often struggle with forming equations from given statements. NCERT solutions break down each question into simple logical steps, helping students clear their doubts effectively.
• Exercise 3.1 introduces students to the concept of linear equations in two variables. NCERT solutions provide step-by-step explanations, helping students grasp the fundamentals clearly.