Pair of Linear Equations in Two Variables

3.1 Introduction

This section introduces the concept of a pair of linear equations in two variables, a fundamental topic in algebra. It begins with a real-life example involving Akhila, who visits a fair and spends money on rides and games. The problem states that the number of times she plays Hoopla is half the number of rides she takes on the Giant Wheel. Each ride costs ₹3 and each Hoopla game costs ₹4, with a total expenditure of ₹20. This situation can be modeled using two variables: x representing the number of rides and y representing the number of Hoopla games played. The relationships are expressed as two linear equations: y = (1/2) x and 3x + 4y = 20. This example illustrates how real-world problems involving two variables and two conditions can be represented as a pair of linear equations. The section sets the stage for exploring various methods to find solutions to such pairs of equations.

• Pair of linear equations involves two linear equations with two variables.
• Real-life situations can be modeled using such equations.
• Variables represent quantities in the problem (e.g., number of rides, number of games).
• Equations express relationships between these variables.
• Solving the pair finds the values of variables satisfying both conditions.
• This chapter explores methods to solve such pairs of equations.
• 📌 Linear equation in two variables: an equation of the form ax + by + c = 0 where a, b, c are constants and x, y are variables.
• 📌 Pair of linear equations: two linear equations involving the same two variables.

3.2 Graphical Method of Solution of a Pair of Linear Equations

This section explains the graphical method to solve a pair of linear equations in two variables. Each linear equation represents a straight line on the Cartesian plane. The solution to the pair corresponds to the point(s) where these lines intersect. The section defines three types of pairs based on their solutions: consistent pairs (with at least one solution) and inconsistent pairs (with no solution). Further, consistent pairs are divided into independent (unique solution) and dependent (infinitely many solutions). The graphical behavior of the lines is summarized as follows: (i) intersecting lines correspond to a unique solution, (ii) parallel lines correspond to no solution, and (iii) coincident lines correspond to infinitely many solutions. Three example pairs illustrate these cases, with their coefficients compared via ratios a1/a2, b1/b2, and c1/c2 to determine the nature of the solution. The section also includes detailed examples solving pairs graphically by plotting points and drawing lines to find the intersection point representing the solution. **Table on page 3 (2×8)** | Sl No. | Pair of lines | $\frac{a_1}{a_2}$ | $\frac{b_1}{b_2}$ | $\frac{c_1}{c_2}$ | Compare the ratios | Graphical representation | Algebraic interpretation | | --- | --- | --- | --- | --- | --- | --- | --- | | 1. | $x - 2y = 0$ **Table on page 4 (5×3)** | x | 0 | 6 | | --- | --- | --- | | y = 6 - x/3 | 2 | 0 | | x | 0 | 3 | | --- | --- | --- | | y = 2x - 12/3 | -4 | -2 | **Table on page 5 (5×3)** | x | 2 | 0 | | --- | --- | --- | | y = 2x - 2 | 2 | -2 | | x | 0 | 1 | | --- | --- | --- | | y = 4x - 4 | -4 | 0 |

• Graph of each linear equation is a straight line on the Cartesian plane.
• Solution of pair corresponds to intersection point(s) of the lines.
• Consistent pair: has at least one solution; inconsistent pair: no solution.
• Dependent pair: infinitely many solutions (coincident lines).
• Ratios of coefficients determine the nature of solution: intersecting, parallel, or coincident.
• Graphical plotting involves finding points satisfying each equation and drawing lines.
• 📌 Consistent pair: pair of linear equations with at least one solution.
• 📌 Inconsistent pair: pair of linear equations with no solution.
• 📌 Dependent pair: pair with infinitely many solutions (coincident lines).