Ncert Class 10th Exercise 3.4

NCERT Class 10th Maths Exercise 3.4: A practical guide

This article provides a detailed explanation and solution to each problem in NCERT Class 10th Mathematics, Exercise 3.4. Even so, this exercise focuses on solving a pair of linear equations in two variables using the substitution method. Understanding this method is crucial for tackling various mathematical problems, and this guide aims to build a solid foundation. We'll break down each question, provide step-by-step solutions, and offer additional tips and tricks to master this crucial concept. This guide ensures you not only understand how to solve these problems but also why the methods work, making you confident in tackling similar questions in the future.

Introduction to the Substitution Method

The substitution method is a powerful algebraic technique used to solve a system of linear equations. A system of linear equations consists of two or more equations, each containing two or more variables. The goal is to find the values of the variables that satisfy all equations simultaneously. The substitution method involves solving one equation for one variable in terms of the other, and then substituting this expression into the second equation to solve for the remaining variable. This process allows us to reduce the system of equations to a single equation with a single variable, which is much easier to solve.

Step-by-Step Guide to Solving Equations using the Substitution Method

The substitution method generally follows these steps:

1. Solve one equation for one variable: Choose one equation and solve it for one variable in terms of the other. As an example, if you have the equations 2x + y = 5 and x - y = 1, you might solve the second equation for x: x = y + 1.

2. Substitute: Substitute the expression you found in step 1 into the other equation. Using our example, substitute (y+1) for x in the first equation: 2(y+1) + y = 5.

3. Solve the resulting equation: This will now be a single equation with only one variable (in our example, y). Solve for this variable.

4. Substitute back: Substitute the value you found in step 3 back into either of the original equations (or the equation from step 1) to solve for the other variable.

5. Check your solution: Substitute both values back into both original equations to verify that they satisfy both equations.

Detailed Solutions to NCERT Class 10th Maths Exercise 3.4

Let's now look at the specific problems of Exercise 3.In practice, 4. Since I do not have access to the specific questions in the NCERT textbook, I will provide a framework and examples demonstrating how to use the substitution method effectively. You can apply this framework to each problem in your textbook.

Example 1:

Solve the following system of equations using the substitution method:

• 3x + y = 11
• x - y = -1

Solution:

1. Solve for one variable: Let's solve the second equation for x: x = y - 1

2. Substitute: Substitute this expression for x into the first equation: 3(y - 1) + y = 11

3. Solve: Expand and simplify: 3y - 3 + y = 11 => 4y = 14 => y = 7/2

4. Substitute back: Substitute y = 7/2 into the equation x = y - 1: x = (7/2) - 1 = 5/2

5. Check: Substitute x = 5/2 and y = 7/2 into both original equations:

   • 3(5/2) + (7/2) = 15/2 + 7/2 = 22/2 = 11 (Correct)
   • (5/2) - (7/2) = -2/2 = -1 (Correct)

That's why, the solution is x = 5/2 and y = 7/2.

Example 2: (Involving fractions)

Solve:

• x/2 + 2y = 8
• 5x - y/2 = 19

Solution:

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1. Solve for one variable: Let's solve the second equation for y: y = 10x - 38

2. Substitute: Substitute this expression for y into the first equation: x/2 + 2(10x - 38) = 8

3. Solve: x/2 + 20x - 76 = 8 => 41x/2 = 84 => x = 168/41

4. Substitute back: Substitute x = 168/41 into y = 10x - 38: y = 10(168/41) - 38 = (1680 - 1558)/41 = 122/41

5. Check: Substitute these values into the original equations to verify (This step is crucial to ensure accuracy).

Example 3: (Involving a variable with a coefficient of 1)

Solve:

• x + 2y = 5
• 2x + 3y = 8

Solution:

1. Solve for one variable: The first equation is easiest to solve for x: x = 5 - 2y

2. Substitute: Substitute this into the second equation: 2(5 - 2y) + 3y = 8

3. Solve: 10 - 4y + 3y = 8 => -y = -2 => y = 2

4. Substitute back: Substitute y = 2 into x = 5 - 2y: x = 5 - 2(2) = 1

5. Check: Verify the solution by substituting x = 1 and y = 2 into both original equations.

Common Mistakes and How to Avoid Them

• Algebraic errors: Pay close attention to the signs when substituting and simplifying equations. Double-check your calculations at each step.
• Incorrect substitution: Ensure you substitute the correct expression into the correct equation.
• Forgetting to check: Always check your solution by substituting the values back into the original equations. This is crucial for catching mistakes.
• Not simplifying correctly: Always simplify the equation before solving for the variable. This reduces the complexity of calculations.

Frequently Asked Questions (FAQ)

• Q: Can I solve the equations in any order? A: Yes, you can solve either equation for either variable first. Still, choosing an equation and variable that simplifies the process is recommended.

• Q: What if I get a solution that doesn't satisfy both equations? A: This indicates an error in your calculations. Carefully review each step of your solution.

• Q: What if the system of equations has no solution or infinitely many solutions? A: In those cases, you'll encounter inconsistencies or identities during the substitution process. These situations imply that the lines represented by the equations are parallel (no solution) or coincide (infinitely many solutions).

Conclusion

Mastering the substitution method for solving a pair of linear equations is a fundamental skill in algebra. Because of that, remember to practice regularly, check your answers, and don't hesitate to seek clarification if you encounter difficulties. 4 and beyond, you can build confidence and proficiency in this essential mathematical technique. With consistent effort, you will master this crucial concept and be well-prepared for more advanced mathematical challenges. Here's the thing — by following the step-by-step process outlined in this guide and practicing with various problems from Exercise 3. Good luck!