ICSE Class 10 Maths: Linear Equations — Methods & Notes 2026

Linear Equations in Two Variables

• ax + by + c = 0; one equation has infinite solutions (a line on graph)
• System of two equations: unique solution (intersecting lines), no solution (parallel), infinite (coincident)
• Graphical method: plot both lines; intersection point = solution

Substitution Method

• Express one variable in terms of the other from one equation
• Substitute in second equation; solve single-variable equation
• Back-substitute to find second variable; check solution in both original equations

Elimination Method

• Multiply equations to make coefficients of one variable equal
• Add or subtract to eliminate that variable; solve for remaining variable
• Preferred when coefficients are integers; error-prone with fractions — use substitution instead

Cross-Multiplication Method

• For a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0:
• x/(b₁c₂−b₂c₁) = y/(c₁a₂−c₂a₁) = 1/(a₁b₂−a₂b₁)
• Quick method when both equations are in standard form

Word Problems — Framing Equations

• Age problems: express ages of two people in terms of variable; use age difference or sum
• Number problems: let units digit = y, tens digit = x; two-digit number = 10x + y
• Speed–time–distance: upstream/downstream problems with boat and current

Consistency and Nature of Solutions

• For unique solution: a₁/a₂ ≠ b₁/b₂
• No solution (inconsistent): a₁/a₂ = b₁/b₂ ≠ c₁/c₂
• Infinite solutions (dependent): a₁/a₂ = b₁/b₂ = c₁/c₂

ICSE Numericals — Linear Equations

• Solve: 3x + 2y = 11 and x − y = 2 by elimination method
• A fraction equals 2/3; if 4 added to numerator and 3 subtracted from denominator, fraction becomes 1 — find it
• Two trains leave simultaneously; set up equations and find meeting point