# NCERT Class 10 Mathematics Chapter 3 Pair of Linear Equations in Two Variables: Summary, Concepts, and Revision Notes | SwaVid This chapter introduces you to pairs of linear equations in two variables, a fundamental concept in algebra. You will learn various methods to solve the... Canonical: https://swavid.com/maths/class/10/chapter/pair-of-linear-equations-in-two-variables Source: https://swavid.com/maths/class/10/chapter/pair-of-linear-equations-in-two-variables # Pair of Linear Equations in Two Variables ## Concepts in this chapter ## The ideas this chapter keeps returning to ## Where this chapter usually trips people up ## Find the gaps before this chapter finds them for you. ## Related chapters ### Pair of Linear Equations in Two Variables Chapter 3 · Class 10 Mathematics This chapter introduces you to pairs of linear equations in two variables, a fundamental concept in algebra. You will learn various methods to solve these systems, both graphically and algebraically, and understand their real-world applications. Your study route Key topics A linear equation in two variables, say x and y, is an equation that can be written in the form ax + by + c = 0, where a, b, and c are real numbers, and a and b are not both zero. A pair of linear equations consists of two such equations. Example Example 1 from Section 3.1: 'Akhila went to a fair in her village. She wanted to enjoy rides on the Giant Wheel and play Hoopla. The number of times she played Hoopla is half the number of rides she had on the Giant Wheel. If each ride costs Rs 3, and a game of Hoopla costs Rs 4, and she spent Rs 20 in total, represent this situation algebraically and graphically.' This example sets up the problem leading to two equations. Watch out Confusing linear equations with non-linear ones (e.g., x^2 + y = 5) or equations with more than two variables. Also, forgetting that 'a' and 'b' cannot both be zero. A linear equation in two variables, say x and y, is an equation that can be written in the form ax + by + c = 0, where a, b, and c are real numbers, and a and b are not both zero. A pair of linear equations consists of two such equations. Tap the card for an example Example Example 1 from Section 3.1: 'Akhila went to a fair in her village. She wanted to enjoy rides on the Giant Wheel and play Hoopla. The number of times she played Hoopla is half the number of rides she had on the Giant Wheel. If each ride costs Rs 3, and a game of Hoopla costs Rs 4, and she spent Rs 20 in total, represent this situation algebraically and graphically.' This example sets up the problem leading to two equations. Why it matters Understanding the basic definition is crucial for correctly identifying and setting up problems that can be solved using these methods. It forms the foundation for all subsequent topics. Watch out Confusing linear equations with non-linear ones (e.g., x^2 + y = 5) or equations with more than two variables. Also, forgetting that 'a' and 'b' cannot both be zero. Ask at home Ask your child to define a linear equation in two variables and provide an example. Then ask them to explain what a 'pair' of such equations means. This chapter delves into pairs of linear equations in two variables, exploring their representation and solution. It begins by defining what constitutes a linear equation in two variables and how a pair of such equations can be represented graphically as two straight lines. Students will learn to interpret the nature of solutions (unique, infinitely many, or no solution) based on whether the lines intersect, coincide, or are parallel. The chapter then introduces three algebraic methods for finding solutions: the substitution method, the elimination method, and the cross-multiplication method. Finally, it covers how certain non-linear equations can be transformed into a pair of linear equations, enabling their solution using the learned techniques. Mastery of these concepts is crucial for solving various real-life problems. Chapter summary This chapter delves into pairs of linear equations in two variables, exploring their representation and solution. It begins by defining what constitutes a linear equation in two variables and how a pair of such equations can be represented graphically as two straight lines. Students will learn to interpret the nature of solutions (unique, infinitely many, or no solution) based on whether the lines intersect, coincide, or are parallel. The chapter then introduces three algebraic methods for finding solutions: the substitution method, the elimination method, and the cross-multiplication method. Finally, it covers how certain non-linear equations can be transformed into a pair of linear equations, enabling their solution using the learned techniques. Mastery of these concepts is crucial for solving various real-life problems. What you should learn Keep these close A pair of linear equations in two variables is of the form a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0. Graphically, a pair of linear equations represents two straight lines, which can intersect (unique solution), be parallel (no solution), or coincide (infinitely many solutions). Conditions for solutions based on coefficients: Unique (a1/a2 ≠ b1/b2), No solution (a1/a2 = b1/b2 ≠ c1/c2), Infinitely many (a1/a2 = b1/b2 = c1/c2). Algebraic methods for solving include Substitution, Elimination, and Cross-Multiplication. Substitution method involves expressing one variable in terms of the other and substituting it into the second equation. Elimination method involves making coefficients of one variable equal and then adding or subtracting the equations. Cross-multiplication method uses a specific formula: x / (b1c2 - b2c1) = y / (c1a2 - c2a1) = 1 / (a1b2 - a2b1). Ensure equations are in standard form (ax + by + c = 0) before applying the cross-multiplication formula. Word problems require careful translation of verbal statements into algebraic equations. Equations that are not linear can sometimes be reduced to linear form by making appropriate substitutions (e.g., p = 1/x, q = 1/y). After solving for substituted variables, remember to find the values of the original variables. Always verify your solutions by substituting them back into the original pair of equations. Common confusions It is easy to think Assuming that all pairs of linear equations will always have a unique solution. The clearer idea Pairs of linear equations can have a unique solution (intersecting lines), no solution (parallel lines), or infinitely many solutions (coincident lines). It's crucial to check the conditions (ratios of coefficients) or the graphical representation. It is easy to think Errors in algebraic manipulation, especially with signs, when using substitution or elimination methods. The clearer idea Pay close attention to positive and negative signs during substitution, multiplication, and addition/subtraction of equations. Double-check each step of algebraic simplification. It is easy to think Incorrectly applying the cross-multiplication formula, especially the order of coefficients or signs. The clearer idea Ensure the equations are first written in the standard form a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0. Carefully memorize the cyclic order of coefficients (b-c-a-b) for the formula and practice its application. It is easy to think Forgetting to find the values of the original variables (x and y) after solving for substituted variables (like p and q) in reducible equations. The clearer idea Always remember that the goal is to find x and y. After solving for p and q, substitute back (e.g., x = 1/p, y = 1/q) to get the final answer. It is easy to think Difficulty in translating word problems into correct algebraic equations. The clearer idea Read the problem carefully, identify the unknown quantities, assign variables (e.g., x and y) to them, and then break down the problem into smaller statements that can be translated into equations one by one. Practice with various types of word problems. Before you push ahead Most stuck chapters trace back to one earlier idea. Check the prerequisites first, or let SwaVid adapt this chapter to the way your child learns. Keep exploring Class 10 Source - Represent a pair of linear equations graphically and interpret the nature of their solutions. - Solve a pair of linear equations in two variables using the substitution method. - Solve a pair of linear equations in two variables using the elimination method. - Solve a pair of linear equations in two variables using the cross-multiplication method. - Formulate and solve word problems by translating them into pairs of linear equations. - Transform equations that are not linear into a pair of linear equations and then solve them. - Represent a pair of linear equations graphically and interpret the nature of their solutions. - Solve a pair of linear equations in two variables using the substitution method. - Solve a pair of linear equations in two variables using the elimination method. - Solve a pair of linear equations in two variables using the cross-multiplication method. - Formulate and solve word problems by translating them into pairs of linear equations. - Transform equations that are not linear into a pair of linear equations and then solve them. - NCERT Class 10 Mathematics textbook: Mathematics : Chapter 3: Pair of Linear Equations in Two Variables ## Key Links - [NCERT Mathematics](https://swavid.com/maths) - [Class 10](https://swavid.com/maths/class/10) - [Start with concepts](https://swavid.com/maths/class/10/chapter/pair-of-linear-equations-in-two-variables) - [Check a prerequisite](https://swavid.com/learning-debt-identifier) - [Concepts](https://swavid.com/maths/class/10/chapter/pair-of-linear-equations-in-two-variables) - [Practice](https://swavid.com/maths/class/10/chapter/pair-of-linear-equations-in-two-variables/practice-questions) - [NCERT Solutions](https://swavid.com/maths/class/10/chapter/pair-of-linear-equations-in-two-variables/ncert-solutions) - [Find the gaps before this chapter](https://swavid.com/learning-debt-identifier) - [Learn this chapter adapted to you](https://swavid.com/learning-style-test) - [Previous chapter 2 . Polynomials](https://swavid.com/maths/class/10/chapter/polynomials) - [Next chapter 4 . Quadratic Equations](https://swavid.com/maths/class/10/chapter/quadratic-equations) - [All Class 10 chapters](https://swavid.com/maths/class/10) - [2 Polynomials Introduction to Polynomials and their Degree · Geometrical Meaning of the Zeroes of a Polynomial Open chapter](https://swavid.com/maths/class/10/chapter/polynomials) - [4 Quadratic Equations Introduction to Quadratic Equations · Solving Quadratic Equations by Factorisation Open chapter](https://swavid.com/maths/class/10/chapter/quadratic-equations) - [1 Real Numbers Euclid's Division Lemma · Euclid's Division Algorithm Open chapter](https://swavid.com/maths/class/10/chapter/real-numbers) - [NCERT Class 10 Mathematics textbook: Mathematics](https://ncert.nic.in/textbook.php)