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Class X Mathematics

Chapter 3: Pair of Linear Equations in Two Variables

Standard NCERT & CBSE aligned study curriculum. Master concepts, track accuracy, revise weak areas, and challenge yourself with 9 customized practice modes.

Chapter Overview

Welcome to Class X Mathematics: Pair of Linear Equations in Two Variables. This chapter forms a core structural component of the math syllabus, designed to build analytical rigor and key formula models.

Use the detailed subtopic guide below to review standard definitions, key mathematical rules, and study guidelines.

Prerequisite Concepts

Linear Equations in Two Variables

Detailed Subtopics Study Guide

Review detailed conceptual explanations, mathematical equations, and guidelines for each subtopic in this chapter:

1Graphical method of solution

Concept Explanation

Solving a system of two linear equations graphically involves plotting both lines on the same coordinate grid. The coordinates of the intersection point (x, y) represent the unique solution to the system.

Mathematical Representation
L_1 \cap L_2 = \{(x_0, y_0)\}
Study Guideline: Find two or three points for each equation, draw the lines, and locate their intersection. Verify the intersection point mathematically by plugging it back into both equations.

2Consistency conditions (Consistent, Inconsistent, Coincident)

Concept Explanation

Consistency conditions determine the number of solutions for a pair of linear equations. A system is consistent (unique solution if lines intersect, infinite solutions if lines coincide) or inconsistent (no solution if lines are parallel).

Mathematical Representation
\text{Unique: } \frac{a_1}{a_2} \neq \frac{b_1}{b_2}, \quad \text{Infinite: } \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}, \quad \text{No Solution: } \frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}
Study Guideline: Compare the ratio of the coefficients of x, y, and the constant terms to determine the nature of solutions without drawing the graph.

3Substitution method

Concept Explanation

The substitution method solves a system of linear equations by expressing one variable in terms of the other from one equation, and then substituting this expression into the second equation to get a single-variable equation.

Mathematical Representation
x = \frac{d - by}{a} \implies \text{Substitute into } a_2x + b_2y + c_2 = 0
Study Guideline: Choose the equation and variable that are easiest to isolate (e.g., a variable with a coefficient of 1) to avoid fractions early in the calculation.

4Elimination method

Concept Explanation

The elimination method solves a system of linear equations by multiplying one or both equations by suitable non-zero constants so that the coefficients of one variable become equal (or opposite). Adding or subtracting the equations then eliminates that variable.

Mathematical Representation
a_1 x + b_1 y = c_1 \, [\times a_2] \, \text{ and } \, a_2 x + b_2 y = c_2 \, [\times a_1] \implies \text{Subtract to eliminate } x
Study Guideline: Multiply the equations so that coefficients of either x or y match. Watch out for signs when subtracting the equations.

5Cross-multiplication method overview

Concept Explanation

The cross-multiplication method is a formula-based algebraic method to solve a pair of linear equations in standard form. It calculates x and y directly using ratios of coefficients.

Mathematical Representation
\frac{x}{b_1c_2 - b_2c_1} = \frac{y}{c_1a_2 - c_2a_1} = \frac{1}{a_1b_2 - a_2b_1}
Study Guideline: Write the coefficients in the pattern: b, c, a, b. Cross-multiply down-wards minus cross-multiplying up-wards to find the denominators.

6Equations reducible to linear form

Concept Explanation

Equations with variables in denominators can be simplified to linear equations using cross-multiplication.

Mathematical Representation
\frac{ax+b}{cx+d} = k \implies ax+b = k(cx+d)
Study Guideline: Multiply the entire equation by the common denominator to clear out fractions.