NCERT Solutions for Class 9 Maths Exercise 4.2 Question 1
Understanding the Question 🧐
This question asks us to determine how many “solutions” a particular linear equation has. A solution is a pair of values for &&x&& and &&y&& that makes the equation true. We need to figure out if there’s only one such pair, exactly two, or an infinite number of pairs for the given equation. This is a core concept covered in the ncert solutions for this chapter.
Which one of the following options is true, and why?
&&y = 3x + 5&& has
- a unique solution
- only two solutions
- infinitely many solutions
Step-by-Step Solution 📝
Let’s investigate the equation &&y = 3x + 5&& by finding some solutions.
Step 1: Find a Few Sample Solutions
We can find a solution by choosing any value for &&x&& and calculating the corresponding value for &&y&&. Let’s try a few simple values for &&x&&.
-
If we choose &&x = 0&&:
&&y = 3(0) + 5&&
&&y = 0 + 5&&
&&y = 5&&
So, &&(0, 5)&& is one solution. -
If we choose &&x = 1&&:
&&y = 3(1) + 5&&
&&y = 3 + 5&&
&&y = 8&&
So, &&(1, 8)&& is another solution. -
If we choose &&x = -1&&:
&&y = 3(-1) + 5&&
&&y = -3 + 5&&
&&y = 2&&
So, &&(-1, 2)&& is a third solution.
Step 2: Evaluate the Options Based on Our Findings
Now let’s look at the given options:
- (i) a unique solution: This is false. We have already found at least three different solutions: &&(0, 5)&&, &&(1, 8)&&, and &&(-1, 2)&&. A unique solution means there is only one.
- (ii) only two solutions: This is also false. We found three solutions, and we could easily find more.
- (iii) infinitely many solutions: This must be true. For any real number we choose for &&x&&, we can perform the calculation &&3x + 5&& to find a corresponding value for &&y&&. Since there is an infinite number of real numbers to choose for &&x&&, there will be an infinite number of solutions.
Conclusion and Justification ✅
The correct option is (iii) infinitely many solutions.
Why? The equation &&y = 3x + 5&& is a linear equation in two variables. The fundamental property of such equations is that for every value of one variable, there is a corresponding value for the other. Since we can substitute an infinite number of values for &&x&& (like &&0, 1, 2, 1.5, \frac{1}{2}, -100&&, etc.), we can generate an infinite number of corresponding &&y&& values, resulting in infinitely many solution pairs &&(x, y)&&.
Think Graphically! 📈
A linear equation in two variables like &&y = 3x + 5&& represents a straight line on a graph. How many points are on a line? Infinitely many! Each point on the line has coordinates &&(x, y)&& that are a solution to the equation.Key Concepts
- A single linear equation in one variable (e.g., &&2x + 4 = 10&&) has a unique solution (in this case, &&x=3&&).
- A single linear equation in two variables (e.g., &&y = 3x + 5&&) has infinitely many solutions.
- A solution is an ordered pair &&(x, y)&& that satisfies the equation.
FAQ (Frequently Asked Questions)
Q: What is the difference between a linear equation in one variable and two variables?
A: A linear equation in one variable (like &&2x + 4 = 10&&) has only one unknown and typically has a unique solution. A linear equation in two variables (like &&y = 3x + 5&&) has two unknowns and has infinitely many solutions, which can be represented as points on a straight line.
Q: Why doesn’t &&y = 3x + 5&& have a unique solution?
A: It doesn’t have a unique solution because there is no restriction on the values that &&x&& or &&y&& can take. You can pick any value for &&x&&, and you will always find a corresponding value for &&y&&. For example, &&(0, 5)&& and &&(1, 8)&& are both solutions. Since we can find more than one, it is not unique.
Q: What does a solution to &&y = 3x + 5&& represent on a graph?
A: Each solution is an ordered pair &&(x, y)&& that represents the coordinates of a point on a Cartesian plane. The collection of all infinite solutions to the equation &&y = 3x + 5&& forms a straight line on the graph.
Q: How can I find a solution to a linear equation in two variables?
A: The easiest way is to choose a simple value for one of the variables (like &&x = 0&& or &&x = 1&&), substitute it into the equation, and then solve for the other variable.
Further Reading
For more information on linear equations and to access the official textbook, you can visit the NCERT website. Official NCERT Website.
