Pair of Linear Equations 📐

Do equations, do variables — graphically, substitution, elimination, aur cross-multiplication.

📖 Introduction — Do equations ek saath

🛒 Real Life Story:

Riya 3 pens aur 2 copies khareedti hai — total ₹70. Sana 1 pen aur 4 copies khareedti hai — total ₹60. Pen aur copy ki alag alag keemat kya hai? Ye solve karne ke liye do equations chahiye — yahi pair of linear equations hai!

x = pen ki keemat, y = copy ki keemat:
3x + 2y = 70 ...(1)
x + 4y = 60 ...(2)

General form: a₁x + b₁y + c₁ = 0 aur a₂x + b₂y + c₂ = 0
Jahan a₁, b₁, a₂, b₂ ≠ 0 (simultaneously)
Graph par har equation ek seedhi line hoti hai.

📊 Graphical Representation

x y Solution (x,y) L₁: a₁x+b₁y=c₁ L₂ Lines intersect at 1 point → Unique solution

🔍 Consistency — 3 Cases

✅ Consistent (Unique)

a₁/a₂ ≠ b₁/b₂
Lines intersect
1 solution

✅ Consistent (Infinite)

a₁/a₂ = b₁/b₂ = c₁/c₂
Lines coincide
∞ solutions

❌ Inconsistent

a₁/a₂ = b₁/b₂ ≠ c₁/c₂
Lines parallel
No solution

ConditionGraphTypeSolutions
a₁/a₂ ≠ b₁/b₂Intersecting linesConsistentExactly 1
a₁/a₂ = b₁/b₂ = c₁/c₂Coincident linesConsistent (dependent)Infinite
a₁/a₂ = b₁/b₂ ≠ c₁/c₂Parallel linesInconsistentNone

🛠️ Method 1: Substitution Method

Ek equation se ek variable ki value nikalo — dusri equation mein substitute karo.
Example: x + y = 5 ...(1) aur 2x − 3y = 4 ...(2)
Step 1
Equation (1) se: x = 5 − y
Step 2
Eq (2) mein substitute: 2(5−y) − 3y = 4 → 10−2y−3y = 4 → −5y = −6 → y = 6/5
Step 3
x = 5 − 6/5 = 19/5
x = 19/5, y = 6/5
Riya-Sana Problem: 3x+2y=70, x+4y=60
From (2): x = 60−4y
In (1): 3(60−4y)+2y = 70 → 180−12y+2y = 70 → −10y = −110 → y = 11
x = 60−44 = 16
Pen = ₹16, Copy = ₹11

🛠️ Method 2: Elimination Method

Dono equations ke coefficients ko multiply karke ek variable eliminate karo.
Example: 3x + 2y = 12 ...(1) aur 2x − y = 1 ...(2)
Multiply (2)×2
4x − 2y = 2 ...(3)
Add (1)+(3)
7x = 14 → x = 2
Substitute x=2 in (1)
6 + 2y = 12 → y = 3
x = 2, y = 3
Example: 5x − 4y + 8 = 0 aur 7x + 6y − 9 = 0
5x − 4y = −8 ...(1)   7x + 6y = 9 ...(2)
(1)×3: 15x − 12y = −24
(2)×2: 14x + 12y = 18
Add: 29x = −6 → x = −6/29
Substitute: y = (5x+8)/4 = (−30/29 + 232/29)/4 = 202/(29×4) = 101/58
(Calculations verify karo substitution se)

🛠️ Method 3: Cross Multiplication

For a₁x + b₁y + c₁=0 aur a₂x + b₂y + c₂=0
x/(b₁c₂ − b₂c₁) = y/(c₁a₂ − c₂a₁) = 1/(a₁b₂ − a₂b₁)
Memory trick — Diagonal cross:
b₁c₂ − b₂c₁  |  c₁a₂ − c₂a₁  |  a₁b₂ − a₂b₁
(b,c) → (c,a) → (a,b) pattern

📝 Word Problems — Strategy

4 Steps:
1. Variables decide karo (x, y kise maanein)
2. Dono conditions se equations banao
3. Solve karo
4. Answer problem ke context mein likhein
Classic: Ek 2-digit number ka sum of digits = 9. Digits palat dene par 27 zyada hota hai. Number nikalo.
Ten's digit = x, Unit's digit = y
x + y = 9 ...(1)
(10y + x) − (10x + y) = 27 → 9y − 9x = 27 → y − x = 3 ...(2)
Add: 2y = 12 → y = 6, x = 3
Number = 36 (Check: 36+27=63 = reversed ✓)
Speed Problem: Train ki distance 300 km. Speed kam ho 100 km/h to 2 ghante late. Speed badhe 100 km/h to 1 ghanta early. Actual speed aur time nikalo.
Let speed = v km/h, time = t hours.
vt = 300
(v−100)(t+2) = 300 → vt + 2v − 100t − 200 = 300 → 2v − 100t = 200 ...(1)
(v+100)(t−1) = 300 → vt − v + 100t − 100 = 300 → −v + 100t = 100 ...(2)
(1)+(2)×2: 2v−100t − 2v + 200t = 200+200 → 100t = 400 → t = 4 hrs
v = 300/4 = 75 km/h... (This approach gives fractions — use in exam per actual NCERT values)

📝 Practice Questions

Q1. Graphically check karo: 2x + 3y = 5 aur 4x + 6y = 10 consistent hai ya nahi?

a₁/a₂ = 2/4 = 1/2; b₁/b₂ = 3/6 = 1/2; c₁/c₂ = 5/10 = 1/2
Sab equal → Lines coincide → Infinitely many solutions (consistent)

Q2. Substitution se solve karo: 7x − 15y = 2 aur x + 2y = 3.

From (2): x = 3−2y. In (1): 7(3−2y)−15y = 2 → 21−14y−15y = 2 → 29y = 19 → y = 19/29
x = 3 − 38/29 = 49/29. x=49/29, y=19/29

Q3. Elimination se: 3x − 5y − 4 = 0 aur 9x = 2y + 7.

3x−5y=4...(1), 9x−2y=7...(2). (1)×3: 9x−15y=12. Subtract (2): −13y=5 → y=−5/13. x=(4+5y)/3=(4−25/13)/3=27/(13×3)=9/13. x=9/13, y=−5/13

Q4. Ek fraction ka numerator denominator se 3 kam hai. Agar 2 dono mein se ghataayein to 1/2 milti hai. Fraction nikalo.

num=x, den=y. x = y−3. (x−2)/(y−2) = 1/2 → 2x−4 = y−2 → 2x−y = 2.
Substitute: 2(y−3)−y = 2 → y = 8, x = 5. Fraction = 5/8

Q5. Kab inconsistent hoga: 2x + ky = 1 aur 3x − 5y = 7?

Inconsistent: a₁/a₂ = b₁/b₂ ≠ c₁/c₂ → 2/3 = k/(−5) → k = −10/3. Check c ratio: 1/7 ≠ 2/3 ✓. k = −10/3
Next: Quadratic Equations →