Linear Equations in Two Variables
About 6 min · Mathematical Reasoning · English
Time worked out from the lesson's 422 words at 150 words a minute, plus a minute for each question.
Linear equations in two variables are part of the SOF IMO syllabus for both Mathematical Reasoning and Achievers sections. A linear equation in two variables takes the form ax + by + c = 0, where a, b, c are real numbers and a, b are not both zero. The solution is always a pair of values (x, y) that satisfies the equation.
Key ideas
- Standard Form: Any linear equation can be written as ax + by + c = 0. Every solution (x, y) of this equation corresponds to a point on a straight line when plotted on the Cartesian plane.
- Graphical Solution: The solution of two linear equations is the point where their graphs intersect. If lines are parallel (no intersection), there's no solution; if they coincide (overlap completely), infinitely many solutions exist.
- Condition for Solutions: For equations a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0, compare ratios: a₁/a₂ ≠ b₁/b₂ gives unique solution; a₁/a₂ = b₁/b₂ ≠ c₁/c₂ gives no solution; a₁/a₂ = b₁/b₂ = c₁/c₂ gives infinite solutions.
- Substitution Method: Express one variable in terms of the other from one equation, then substitute into the second equation to get a single-variable equation. Most efficient when one coefficient is already 1 or when one equation is simpler.
- Elimination Method: Multiply equations by suitable numbers to make coefficients of one variable equal (or opposite), then add or subtract equations to eliminate that variable. Best when coefficients can be easily matched.
Quick reference
- Three solution methods: substitution (simplest equation), elimination (equal/opposite coefficients), cross-multiplication (standard form equations)
- Check solution type first: compare a₁/a₂ vs b₁/b₂ to predict unique/no/infinite solutions before solving
- Graphical solution = intersection point of two lines; parallel lines = no solution; overlapping lines = infinite solutions
- Always verify your answer by substituting back into both original equations
- Word problems: define variables clearly, form two separate equations from two different conditions, solve, check practicality
Check yourself: 3 questions
1. Solve the following pair of linear equations by substitution method: x + y = 14 x - y = 4 What is the value of x?
2. A fraction becomes 1/3 when 1 is subtracted from the numerator and it becomes 1/4 when 8 is added to its denominator. Find the fraction.
3. For what value of k will the following pair of linear equations have infinitely many solutions? 2x + 3y = 7 (k + 2)x + (2k + 1)y = 3(2k - 1)
This lesson is cut from the topic's note, not rewritten. Fact-checked on 21 Oct 2026: an AI-assisted check of this note against cited sources; the errors it found were corrected that day. Some errors may remain. The questions are from Shishya's validated question bank.
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