Cheat Sheet — Pair of Linear Equations

📐 Standard Form

General Form

a₁x + b₁y + c₁ = 0
a₂x + b₂y + c₂ = 0
where a² + b² ≠ 0 for each equation

Derived Quantities (b ≠ 0)

Slope = −a/b
Y-intercept = −c/b
X-intercept = −c/a

SolutionA pair (x₀, y₀) satisfying BOTH equations simultaneously

Solution set0, 1, or ∞ solutions — never exactly 2

Convert tip: Move all terms left, set = 0, then read off a, b, c.

📈 Graphical Method

Three Cases

Graph	Condition	Solutions	System
Lines intersect	a₁/a₂ ≠ b₁/b₂	Unique	Consistent
Lines parallel	a₁/a₂ = b₁/b₂ ≠ c₁/c₂	None	Inconsistent
Coincident lines	a₁/a₂ = b₁/b₂ = c₁/c₂	Infinitely many	Consistent

Plotting steps: Set x=0 → find y; set y=0 → find x. Plot both points, draw line.

Limitation: Graphical method gives approximate answers for non-integer solutions.

🔢 Consistency Conditions

The Master Table

Condition	Type	Geometric
a₁/a₂ ≠ b₁/b₂	Unique solution ✓	Intersecting
a₁/a₂ = b₁/b₂ ≠ c₁/c₂	No solution ✗	Parallel
a₁/a₂ = b₁/b₂ = c₁/c₂	Infinite solutions ∞	Coincident

Quick Algorithm

Compute ratio r₁ = a₁/a₂ and r₂ = b₁/b₂
If r₁ ≠ r₂ → unique solution
If r₁ = r₂, compute r₃ = c₁/c₂
r₁ = r₂ ≠ r₃ → no solution; r₁ = r₂ = r₃ → infinite

🔄 Substitution Method

Steps

From the simpler equation, express one variable in terms of the other (choose coefficient 1 or −1 to avoid fractions)
Substitute the expression into the other equation
Solve the resulting single-variable equation
Back-substitute to find the second variable
Verify in BOTH original equations

Worked Example

x − y = 1 … (1)
2x + y = 8 … (2)
From (1): x = y + 1
Sub in (2): 2(y+1) + y = 8
3y = 6 → y = 2
Back-sub: x = 3 ✓

Best when: One equation has a variable with coefficient ±1.

✂️ Elimination Method

Steps

Multiply equations by suitable multipliers to make one variable's coefficients equal in magnitude
Add equations if signs of equal coefficients are opposite; subtract if same sign
Solve the resulting single-variable equation
Substitute back to find the other variable
Verify in BOTH original equations

Worked Example

3x + 2y = 16 … (1)
5x − 2y = 8 … (2)
Coefficients of y: +2 and −2 (opposite)
ADD: 8x = 24 → x = 3
Sub in (1): y = 7/2 = 3.5 ✓

Best when: Coefficients share an LCM easily, or are already equal.

✖️ Cross-Multiplication Method

Formula (equations must be in ax + by + c = 0 form)

x/(b₁c₂ − b₂c₁) = y/(c₁a₂ − c₂a₁) = 1/(a₁b₂ − a₂b₁)

Butterfly Diagram

b₁✕c₁✕a₁✕b₁

cross products (each row vs next)

b₂✕c₂✕a₂✕b₂

          x-den: b₁c₂−b₂c₁  |  y-den: c₁a₂−c₂a₁  |  divisor: a₁b₂−a₂b₁
        

Remember: Equations MUST be in ax + by + c = 0 form (not = k). Fails if a₁b₂ − a₂b₁ = 0.

🎯 Word Problem Types

Type	Variables	Key Equations
Age	x = older, y = younger	Present relation + future/past relation
Money	x = cost₁, y = cost₂	Two purchase scenarios
Digit	a = tens, b = units	Number = 10a+b; reversed = 10b+a
Speed / Stream	u = boat, v = stream	Down: u+v; Up: u−v; D = S×T
Fraction	x = num, y = den	Two conditions after cross-multiply
Geometry	angles / sides	Angle sum + given relationship

Strategy: Define → Translate → Solve → Verify in original problem context.

🧠 Memory Tricks & Exam Tips

Remember the Ratio Rule

"IPC" Memory:
Intersecting → 1 solution (a₁/a₂ ≠ b₁/b₂)
Parallel → 0 solutions (a₁/a₂ = b₁/b₂ ≠ c₁/c₂)
Coincident → ∞ solutions (all 3 ratios equal)

Common Errors

Forgetting to move c to left: In cross-multiplication, c₁ is from ax + by + c = 0, so if equation is 2x + 3y = 7, then c = −7, not +7.

Back-substitution error: Always verify in BOTH equations, not just the one you used.

Method Selection Guide

Situation	Best Method
One variable has coefficient ±1	Substitution
Coefficients share easy LCM	Elimination
Quick/formulaic answer needed	Cross-multiplication
Approximate or visual answer OK	Graphical

Exam tip: For 3-mark "solve the pair" questions, always write verification at the end — it earns the final mark.