NCERT Solutions for Class 9 Maths Exercise 4.1 Question 1
Understanding the Question 🧐
This question asks us to translate a real-world statement into a mathematical equation. Specifically, we need to create a linear equation in two variables. This is a fundamental skill in algebra where we represent relationships using variables.
The cost of a notebook is twice the cost of a pen. Write a linear equation in two variables to represent this statement.
(Hint: Take the cost of a notebook to be &&₹\ x&& and the cost of a pen to be &&₹\ y&&).
Step-by-Step Solution 📝
Let’s break down the problem into simple, easy-to-follow steps to get our final answer.
Step 1: Define the Variables
The first step in any word problem is to identify the unknown quantities and assign variables to them. The question kindly gives us a hint for this!
- Let the cost of one notebook be &&₹\ x&&.
- Let the cost of one pen be &&₹\ y&&.
Step 2: Translate the Statement into a Mathematical Relationship
Now, we’ll convert the given sentence, “The cost of a notebook is twice the cost of a pen,” into an equation using our variables.
- “The cost of a notebook” is represented by &&x&&.
- The word “is” translates to the equals sign (&&=&&).
- The word “twice” means to multiply by 2 (&&2 \times&&).
- “the cost of a pen” is represented by &&y&&.
Putting it all together, we get:
(Cost of a notebook) = 2 &&\times&& (Cost of a pen)
Step 3: Form the Equation
Substituting the variables into the relationship from Step 2, we get our initial equation.
&&x = 2y&&
Step 4: Write the Equation in Standard Form
A linear equation in two variables is usually written in the standard form &&ax + by + c = 0&&. To do this, we need to move all the terms to one side of the equation.
We have the equation &&x = 2y&&.
To bring &&2y&& to the left side, we subtract &&2y&& from both sides:
&&x – 2y = 2y – 2y&&
&&x – 2y = 0&&
This is the required linear equation in two variables.
Conclusion and Key Points ✅
The linear equation in two variables that represents the statement “The cost of a notebook is twice the cost of a pen” is:
Points to Remember
- A linear equation in two variables has the standard form &&ax + by + c = 0&&.
- In our final equation, &&x – 2y = 0&&, the coefficients are &&a = 1&&, &&b = -2&&, and the constant is &&c = 0&&.
- Translating words into mathematical symbols is a key skill: ‘is’ often means ‘=’, ‘twice’ means ‘&&2 \times&&’, ‘more than’ means ‘+’, etc.
FAQ (Frequently Asked Questions)
Q: What is a linear equation in two variables?
A: A linear equation in two variables is an equation that can be written in the standard form &&ax + by + c = 0&&, where &&x&& and &&y&& are variables, and &&a&&, &&b&&, and &&c&& are real numbers, with the condition that &&a&& and &&b&& are not both zero. Its graph is always a straight line.
Q: How do you translate ‘the cost of a notebook is twice the cost of a pen’ into an equation?
A: First, assign variables: let the cost of the notebook be &&x&& and the cost of the pen be &&y&&. The word ‘is’ translates to &&=&&, and ‘twice’ means multiplying by &&2&&. So, the statement becomes ‘cost of notebook’ &&= 2 \times&& ‘cost of pen’, or &&x = 2y&&.
Q: What is the standard form of the equation for this question?
A: The initial equation is &&x = 2y&&. To write it in the standard form (&&ax + by + c = 0&&), you move all terms to one side. Subtracting &&2y&& from both sides gives the standard form: &&x – 2y = 0&&.
Q: In the equation &&x – 2y = 0&&, what are the values of &&a, b,&& and &&c&&?
A: Comparing &&x – 2y = 0&& with the standard form &&ax + by + c = 0&&, we can identify the coefficients: &&a = 1&&, &&b = -2&&, and &&c = 0&&.
Q: Is &&2y – x = 0&& also a correct answer?
A: Yes, &&2y – x = 0&& is also a correct representation. It is equivalent to &&x – 2y = 0&&, as you can get one from the other by multiplying the entire equation by &&-1&&.
Further Reading
For more information on linear equations and to access the official textbook, you can visit the NCERT website. Official NCERT Website.
