In Class 10 Maths, the chapter on Pair of Linear Equations in Two Variables is crucial for building problem-solving skills and understanding algebraic relationships. This chapter teaches students how to solve two linear equations simultaneously, which is an essential skill for various real-life applications. The methods used to solve these equations include graphical, substitution, elimination, and cross-multiplication methods. These approaches provide different perspectives, enabling students to find the values of the variables that satisfy both equations.
The NCERT solutions for Class 10 Maths provide detailed, step-by-step explanations for each method, helping students grasp the concepts with ease. Each solution is designed to give clarity and boost confidence in solving equations. Whether you're preparing for your board exams or just looking to master this fundamental topic, the NCERT solutions serve as an invaluable resource.
Graphical representation, for example, allows students to visualize the solutions by plotting the equations on a graph. On the other hand, algebraic methods like substitution and elimination provide more efficient ways to arrive at solutions without relying on a graph. The cross-multiplication method, though less common, offers a direct formulaic approach to solving the equations.
Understanding how to solve pair of linear equations lays the foundation for higher-level mathematics, particularly in topics like matrices, determinants, and calculus. With the NCERT solutions, students can improve their problem-solving techniques and approach challenging questions with confidence. By practicing these methods, students can tackle any linear equation problem and apply their knowledge to practical scenarios.
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Exercise 3.1 - NCERT Solutions
In Exercise 3.1 of NCERT Class 10 Maths, students are required to solve a pair of linear equations in two variables using the substitution method. This exercise helps students understand how to isolate one variable in terms of the other, and then substitute it into the second equation to find the value of the unknowns. The substitution method is one of the simplest ways to solve linear equations, especially when one equation can be easily solved for one of the variables.
In this exercise, students will practice solving equations such as ax + by = c and dx + ey = f by substituting one of the variables. This method is particularly useful when one equation is easy to solve for a single variable, allowing students to reduce the complexity of the system of equations. As students progress through the exercise, they will gain confidence in handling more complicated systems of linear equations.
The step-by-step approach in the NCERT solutions ensures that students learn how to systematically approach such problems. Each solution includes detailed explanations of how to isolate a variable, substitute it, and then solve for the remaining variable. This reinforces both conceptual understanding and problem-solving skills. It is recommended that students practice solving these types of equations to master the method and improve their exam preparation.
To enhance your learning, watch this video which demonstrates the substitution method in detail:
Exercise 3.2 - NCERT Solutions
Exercise 3.2 of NCERT Class 10 Maths focuses on solving a pair of linear equations using the elimination method. This method involves eliminating one of the variables by adding or subtracting the equations. The goal is to simplify the system of equations to a single variable, which can then be solved easily. This approach is particularly effective when the coefficients of one of the variables are the same or easily made the same by multiplying both equations by appropriate numbers.
In this exercise, students are given two equations of the form ax + by = c and dx + ey = f, and they must manipulate the equations to eliminate one of the variables. This method is often preferred when dealing with larger or more complex systems of equations, as it avoids the need to isolate variables and substitute them. The elimination method provides a quicker way to solve linear equations in many cases.
The NCERT solutions for this exercise walk students through every step, from multiplying the equations to cancel out one variable, to solving for the remaining variable. This systematic approach helps students gain a strong grasp of how to use the elimination method efficiently. By practicing these problems, students can enhance their problem-solving ability and tackle even the most challenging linear equations with confidence.
To further understand the elimination method, watch this video:
Exercise 3.3 - NCERT Solutions
Exercise 3.3 of Class 10 NCERT Maths introduces word problems based on linear equations in two variables. This is one of the most critical exercises in the chapter as it helps students understand how to convert real-life scenarios into mathematical equations. These problems often include scenarios like age-related puzzles, motion problems, financial transactions, and mixtures, which help students apply algebra in practical contexts.
To solve these problems, students must read the question carefully, identify the two variables involved, and then form two linear equations based on the given conditions. After that, they can use any of the three methods — substitution, elimination, or cross-multiplication — to solve the equations. This exercise tests a student's comprehension skills along with their algebraic abilities.
The challenge here is not just solving equations, but correctly translating language-based questions into equation form. That makes this exercise particularly important from an exam perspective. Most CBSE board papers pick at least one question from this exercise due to its real-life application and problem-solving depth.
To understand how to tackle these word problems, check out this video solution:
Exercise 3.4 - NCERT Solutions
Exercise 3.4 focuses on solving linear equations using the cross-multiplication method. This method is algebraically advanced and efficient, especially for equations in the standard form. In this method, the system of equations is rearranged to the form: ax + by = c and dx + ey = f. Students then apply the cross multiplication technique to find x and y directly using a formula.
This method involves cross-multiplying the coefficients in a specific order and then dividing accordingly to get the values of x and y. While this method avoids the long steps of elimination or substitution, it requires students to handle the coefficients carefully to avoid calculation errors. Many students prefer it because it offers a shortcut, especially in objective-type questions.
The NCERT solutions guide students step-by-step to ensure that the technique is grasped conceptually, not just mechanically. Students also get to practice simplifying their answers properly, which is a useful skill in exams. Mastering this method can improve accuracy and speed in solving problems.
Watch this helpful explanation video on cross-multiplication for this exercise:
Exercise 3.5 - NCERT Solutions
Exercise 3.5 wraps up the chapter with a mix of conceptual problems, word problems, and revision questions involving all three major solving techniques: substitution, elimination, and cross multiplication. It tests a student’s understanding of when to use which method effectively and how to analyze a problem quickly to identify the best approach.
The questions in this exercise often involve unique conditions such as “infinitely many solutions” or “no solution,” based on the consistency of equations. It also includes real-life-based problem sets like those related to trains, coins, and ages. This makes it an ideal revision tool to prepare for final exams and test applications from all angles.
NCERT solutions help students revise the entire chapter through Exercise 3.5 by providing clear, step-by-step solutions for each question. Students are advised to attempt this exercise after thoroughly understanding the previous ones, as it acts as a summary and consolidation point for the chapter.
For complete clarity on the questions and methods in Exercise 3.5, watch this video solution:
Graphical Method
The graphical method is a visual approach to solving a pair of linear equations. In this method, the two equations are represented as straight lines on a coordinate plane, and the point where the two lines intersect gives the solution to the system of equations. This method is helpful in understanding the geometric representation of the equations and provides a clear visual insight into how the two variables relate to each other.
In this method, the general form of a linear equation is ax + by = c. To plot the equation on a graph, we first need to rewrite it in slope-intercept form or calculate two points that satisfy the equation. Once both lines are plotted, the point of intersection is the solution. If the lines are parallel, the system has no solution; if they coincide, there are infinitely many solutions.
The graphical method is simple and intuitive, making it easy for students to understand the relationship between the two equations. However, it may not be as precise when dealing with large numbers or when the solution lies between grid points. Despite these limitations, this method provides a strong conceptual foundation for understanding linear equations and their solutions.
To see how the graphical method works, watch this video:
Substitution Method
The substitution method is an algebraic approach to solving linear equations in two variables. In this method, one equation is solved for one variable in terms of the other, and this expression is then substituted into the second equation. The goal is to reduce the system to a single-variable equation, which can be solved easily to find the value of one variable. Once the value of one variable is determined, it is substituted back into one of the original equations to find the value of the other variable.
This method is especially useful when one of the equations can be easily manipulated to isolate a variable. For example, if one of the equations is in the form x = ..., it is simple to substitute this expression into the second equation. The substitution method is systematic and efficient when the equations are easily solvable for one variable.
The main advantage of the substitution method is its flexibility and ease of use, especially when dealing with simple linear equations. However, it can become cumbersome with more complex equations or systems with fractional coefficients. Nevertheless, this method is a fundamental technique that students must master for solving linear systems.
For a better understanding, watch this video on substitution method:
Elimination Method
The elimination method is another popular algebraic technique used to solve a pair of linear equations. In this method, the goal is to eliminate one of the variables by adding or subtracting the two equations. This is typically done by making the coefficients of one of the variables the same in both equations. Once one variable is eliminated, the remaining equation is easier to solve for the other variable.
The elimination method works best when both equations have terms with the same or easily adjustable coefficients. Students multiply both equations by appropriate numbers to equalize the coefficients of one variable, and then add or subtract the equations to eliminate that variable. The method is efficient and often faster than substitution, especially when the equations are designed to have easily relatable coefficients.
Although the elimination method is generally faster and more straightforward than the substitution method, it may require more algebraic manipulation. It is a useful tool in solving linear systems and is a vital part of any student’s toolkit for solving algebraic equations.
Watch this video to learn more about the elimination method:
Cross Multiplication Method
The cross multiplication method is a less commonly used but highly efficient technique for solving a pair of linear equations. In this method, the two equations are written in the form ax + by = c and dx + ey = f, and then the solution is derived by cross multiplying the coefficients. This method results in a quick, formula-based solution for finding the values of the two variables.
To use the cross multiplication method, students multiply the coefficients and constants in a specific pattern. The resulting formulas allow students to directly solve for both variables. The method is particularly useful when students need a quick solution and when the equations are set up in a way that makes cross multiplication easy.
The cross multiplication method is less visual than the graphical method but is often preferred for its directness. Although it is not as widely used as the substitution or elimination methods, it is an important technique to master, particularly for its efficiency in solving equations.
Learn more about the cross multiplication method in this video:
Frequently Asked Questions (FAQs)
What are the main methods to solve a pair of linear equations?
The main methods include graphical, substitution, elimination, and cross-multiplication methods. Each has its own benefits depending on the type of equation.
Which exercise focuses on word problems?
Exercise 3.3 in NCERT Class 10 Maths focuses entirely on real-life based word problems involving linear equations in two variables.
How to check if a pair of equations has infinite solutions?
Compare the ratios: if a₁/a₂ = b₁/b₂ = c₁/c₂, the system has infinitely many solutions. This is covered in Exercise 3.5.
Is cross multiplication useful for solving quickly?
Yes, it offers a direct way to solve two-variable equations without substituting or eliminating terms. It's useful in objective-type questions.
Where can I get all Class 10 Maths formulas chapterwise?
You can refer to this complete resource: Class 10 Maths All Formulas Chapterwise .