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Schooling · Class 10 · Mathematics · Pair of Linear Equations in Two Variables · Chapter on a page

CBSE Class 10 · Mathematics · NCERT Mathematics

Pair of Linear Equations in Two Variables: chapter on a page

A cut of Shishya's notes for this chapter: the first paragraph of the introduction and the key ideas, formulas and facts, common mistakes and quick revision lists, word for word. Nothing is added or reworded; the worked examples are left out (they are on the chapter page).

About the notes: Written by Shishya's AI on 26 Sept 2026 from the chapter's title and class level, in Shishya's own words — not a copy or summary of the textbook. Read the official chapter for the book's own text, activities and exercises.

This note's current text has no applied fact-check on record.

Mind map

Drawn from the same notes by a fixed rule (no AI): each label is the item's own words — its bold term, its label before a colon, a short formula as written, or its first words.

Mind map of Pair of Linear Equations in Two Variables: Key ideas, Formulas and facts, Watch out forKey ideasA linear equation in two variables…A pair of linear equations…If two lines intersect at exactly…For equations a₁x + b₁y + c₁ = 0 and…In substitution method, express one…In elimination method, multiply…A consistent pair has at least one…Formulas and factsFor the pair a₁x + b₁y + c₁ = 0 and…If a₁/a₂ ≠ b₁/b₂, lines intersectIf a₁/a₂ = b₁/b₂ ≠ c₁/c₂If a₁/a₂ = b₁/b₂ = c₁/c₂SubstitutionEliminationWatch out forForgetting to change signs when…Substituting incorrectly by mixing…Confusing the ratio conditionsDropping a negative sign when…Stopping after finding one variablePair of LinearEquations in TwoVariables
The mind map as a list
  • Key ideas
    • A linear equation in two variables…
    • A pair of linear equations…
    • If two lines intersect at exactly…
    • For equations a₁x + b₁y + c₁ = 0 and…
    • In substitution method, express one…
    • In elimination method, multiply…
    • A consistent pair has at least one…
  • Formulas and facts
    • For the pair a₁x + b₁y + c₁ = 0 and…
    • If a₁/a₂ ≠ b₁/b₂, lines intersect
    • If a₁/a₂ = b₁/b₂ ≠ c₁/c₂
    • If a₁/a₂ = b₁/b₂ = c₁/c₂
    • Substitution
    • Elimination
  • Watch out for
    • Forgetting to change signs when…
    • Substituting incorrectly by mixing…
    • Confusing the ratio conditions
    • Dropping a negative sign when…
    • Stopping after finding one variable

What this chapter is about

This chapter teaches you how to work with two linear equations that share two unknowns. In earlier classes, you solved single linear equations in one variable. Now you learn to handle situations where two conditions must be satisfied at the same time, leading to two equations that must be solved together.

Key ideas

  • A linear equation in two variables has the form ax + by + c = 0, where a, b, c are real numbers and at least one of a or b is non-zero. Its graph is a straight line.
  • A pair of linear equations represents two straight lines. The solution of the pair is a point (x, y) that satisfies both equations simultaneously.
  • If two lines intersect at exactly one point, the pair has a unique solution; if they are parallel, the pair has no solution; if they coincide (same line), the pair has infinitely many solutions.
  • For equations a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0, compare the ratios a₁/a₂, b₁/b₂ and c₁/c₂ to decide the type of solution without solving.
  • In substitution method, express one variable in terms of the other from one equation, then substitute into the second equation.
  • In elimination method, multiply equations by suitable numbers so that one variable has equal (or opposite) coefficients, then add or subtract to eliminate that variable.
  • A consistent pair has at least one solution; an inconsistent pair has no solution.

Formulas and facts to remember

  • For the pair a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0:
  • If a₁/a₂ ≠ b₁/b₂, lines intersect → exactly one solution (consistent).
  • If a₁/a₂ = b₁/b₂ ≠ c₁/c₂, lines are parallel → no solution (inconsistent).
  • If a₁/a₂ = b₁/b₂ = c₁/c₂, lines coincide → infinitely many solutions (consistent, dependent).
  • Substitution: From one equation, get y = (expression in x), put it in the other equation, solve for x, then find y.
  • Elimination: Make coefficients of one variable same in magnitude, add or subtract equations, solve for the remaining variable.

Common mistakes

  • Forgetting to change signs when subtracting entire equations → write out the subtraction term by term before simplifying.
  • Substituting incorrectly by mixing up which variable was expressed → always replace the exact expression you derived.
  • Confusing the ratio conditions: thinking equal first two ratios always means no solution → check the third ratio too; equal all three means infinitely many solutions.
  • Dropping a negative sign when clearing fractions → multiply every term, including constants, on both sides.
  • Stopping after finding one variable → always substitute back to find the second variable and verify in both original equations.

Quick revision

  • Two linear equations in x and y can have one solution, no solution, or infinitely many solutions.
  • Compare a₁/a₂ with b₁/b₂ and c₁/c₂ to predict the type of solution without solving.
  • Substitution works well when one equation easily gives a variable in terms of the other.
  • Elimination works well when coefficients can be matched by simple multiplication.
  • Always verify your answer by substituting both values into both original equations.