Linear Equations in Two Variables — Study Notes

Overview

Linear equations in two variables form the backbone of algebraic problem solving in SOF IMO and are crucial 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.

This topic is heavily tested through graphical interpretation questions, word problems requiring equation formulation, and direct solution problems using three standard methods. Students must master both the mechanical solving techniques and the graphical understanding—many IMO questions test whether you can visualize the solution geometrically or choose the most efficient solution method. The topic connects to coordinate geometry, ratio-proportion problems, and real-world applications like mixture problems and age-related puzzles.

Understanding when a pair of linear equations has a unique solution, infinitely many solutions, or no solution is particularly important for competitive scenarios. This concept appears in both direct questions and as part of multi-step problems in the Achievers section.

Key Concepts

• **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.

• **Cross-Multiplication Method**: Direct formula-based approach where x/(b₁c₂ - b₂c₁) = y/(c₁a₂ - c₂a₁) = 1/(a₁b₂ - a₂b₁). Fastest for equations already in standard form with no zero coefficients.

• **Word Problem Strategy**: Identify two unknowns, form two equations from given conditions, solve using any method, then verify if the solution makes practical sense in context.

• **Consistency Check**: After solving, always substitute your answer back into both original equations to verify correctness—critical for avoiding sign errors and calculation mistakes.

Formulas / Key Facts

**Cross-Multiplication Formula**: For a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0, x = (b₁c₂ - b₂c₁)/(a₁b₂ - a₂b₁) and y = (c₁a₂ - c₂a₁)/(a₁b₂ - a₂b₁)

**Number of Solutions**:

• Unique solution: a₁/a₂ ≠ b₁/b₂ (intersecting lines)
• No solution: a₁/a₂ = b₁/b₂ ≠ c₁/c₂ (parallel lines)
• Infinite solutions: a₁/a₂ = b₁/b₂ = c₁/c₂ (coincident lines)

**Standard Form Conversion**: For equation y = mx + c, standard form is mx - y + c = 0

**Point of Intersection**: For lines with slopes m₁ and m₂, they intersect if m₁ ≠ m₂

**Solution Pair Check**: (x, y) is a solution if it satisfies both equations simultaneously

**Elimination Multiplier Rule**: To eliminate x, multiply first equation by a₂ and second by a₁, then subtract

**Substitution Isolation**: From ax + by = c, express x = (c - by)/a before substituting

Worked Examples

**Example 1**: Solve by substitution: 2x + 3y = 11 and 3x - y = 5

Step 1: From second equation, express y in terms of x: y = 3x - 5

Step 2: Substitute into first equation: 2x + 3(3x - 5) = 11

Step 3: Simplify: 2x + 9x - 15 = 11 → 11x = 26 → x = 26/11

Step 4: Substitute x back: y = 3(26/11) - 5 = 78/11 - 55/11 = 23/11

Solution: x = 26/11, y = 23/11

**Example 2**: Solve by elimination: 5x + 2y = 16 and 3x + 2y = 10

Step 1: Notice y coefficients are already equal (both 2)

Step 2: Subtract second equation from first: (5x + 2y) - (3x + 2y) = 16 - 10

Step 3: Simplify: 2x = 6 → x = 3

Step 4: Substitute x = 3 into first equation: 5(3) + 2y = 16 → 15 + 2y = 16 → y = 0.5

Solution: x = 3, y = 0.5

**Example 3**: Solve by cross-multiplication: 2x + 3y - 7 = 0 and 3x - 2y - 4 = 0

Step 1: Identify coefficients: a₁ = 2, b₁ = 3, c₁ = -7; a₂ = 3, b₂ = -2, c₂ = -4

Step 2: Apply formula for x: x = (b₁c₂ - b₂c₁)/(a₁b₂ - a₂b₁) = (3×(-4) - (-2)×(-7))/(2×(-2) - 3×3) = (-12 - 14)/(-4 - 9) = -26/(-13) = 2

Step 3: Apply formula for y: y = (c₁a₂ - c₂a₁)/(a₁b₂ - a₂b₁) = ((-7)×3 - (-4)×2)/(-13) = (-21 + 8)/(-13) = -13/(-13) = 1

Solution: x = 2, y = 1

Common Mistakes

**Sign Error in Elimination**: Students often make mistakes with signs when subtracting equations. **Fix**: Write out the subtraction explicitly as (term₁ - term₂) for each position before simplifying. When eliminating by addition, make one coefficient negative first.

**Wrong Variable Substitution**: Substituting incorrectly by forgetting to distribute or handle brackets properly. **Fix**: Always write the complete substitution step showing what expression replaces the variable, then expand brackets carefully before collecting terms.

**Ratio Comparison Errors**: Confusing the condition checks by comparing wrong ratios or forgetting to simplify fractions. **Fix**: Always simplify each ratio a₁/a₂, b₁/b₂, c₁/c₂ completely before comparing. Write them side-by-side to avoid confusion.

**Cross-Multiplication Formula Mix-up**: Reversing numerator expressions or denominator calculations. **Fix**: Memorize using the diagonal pattern: for x numerator, cross b₁ with c₂ and b₂ with c₁. Denominator is always a₁b₂ - a₂b₁ for both x and y.

**Forgetting to Verify Solutions**: Calculating an answer but not checking if it satisfies both original equations. **Fix**: Make verification a mandatory final step. Substitute your (x, y) values into both equations and confirm both yield true statements.

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
• Cross-multiplication denominator (a₁b₂ - a₂b₁) must never be zero for unique solution to exist