NCERT Solutions for Class 9 Maths Exercise 4.2 Question 4
Understanding the Question 🧐
This question gives us a linear equation with an unknown constant, &&k&&. It also gives us a specific pair of values for &&x&& and &&y&& and tells us that this pair is a “solution” to the equation. Our task is to use this information to find the value of &&k&&. The key idea here, as you’ll see in these ncert solutions, is that if a point is a solution, it must make the equation true.
Find the value of &&k&&, if &&x = 2, y = 1&& is a solution of the equation &&2x + 3y = k&&.
Step-by-Step Solution 📝
Let’s break down the problem into simple, logical steps to find the value of &&k&&.
Step 1: Identify the Given Information
First, let’s write down what we know from the question.
- The equation is: &&2x + 3y = k&&
- A known solution is: &&x = 2&& and &&y = 1&&
Step 2: Use the Definition of a Solution
The statement “&&x = 2, y = 1&& is a solution” is the most important clue. It means that if we substitute these specific values of &&x&& and &&y&& into the equation, the equation must hold true.
Step 3: Substitute the Values into the Equation
We will now replace &&x&& with &&2&& and &&y&& with &&1&& in the equation &&2x + 3y = k&&.
&&2(2) + 3(1) = k&&
Step 4: Calculate and Solve for &&k&&
Now, we just need to simplify the left-hand side of the equation to find our answer.
&&4 + 3 = k&&
&&7 = k&&
Conclusion and Final Answer ✅
By substituting the given solution into the equation, we found the value of the unknown constant.
Therefore, the value of &&k&& is &&7&&.
The complete equation is &&2x + 3y = 7&&.
Key Insight
This question might seem tricky, but it’s just a simple substitution problem in disguise! When you’re told something ‘is a solution,’ it’s a direct instruction to plug those values into the equation.Core Concepts
- If a point &&(a, b)&& is a solution to an equation, it will make the equation true when you set &&x = a&& and &&y = b&&.
- This substitution method is a common and powerful technique for finding unknown constants in equations.
FAQ (Frequently Asked Questions)
Q: What does it mean when a point is a “solution” to an equation?
A: When a point &&(x, y)&& is a solution to an equation, it means that if you substitute the values of &&x&& and &&y&& from that point into the equation, the equation will be true (the left side will equal the right side).
Q: What is the first step to find an unknown like &&k&& when given a solution?
A: The very first step is to substitute the given &&x&& and &&y&& values from the solution directly into the equation.
Q: What is the complete equation after finding &&k&&?
A: After finding that &&k = 7&&, the complete equation is &&2x + 3y = 7&&.
Q: Could this equation have other solutions besides &&(2, 1)&&?
A: Yes, absolutely. The equation &&2x + 3y = 7&& is a linear equation in two variables, so it has infinitely many solutions. The point &&(2, 1)&& is just one of them. For example, &&(x=-1, y=3)&& is another solution.
Further Reading
For more information on linear equations and to access the official textbook, you can visit the NCERT website. Official NCERT Website.
