Pair of Linear Equations – Elimination Method | Class 10 Maths Complete Guide

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Pair of Linear Equations – Elimination Method Kya Hai?

Class 10 Maths me do linear equations with two variables ko Pair of Linear Equations kaha jata hai. In equations me generally do variables hote hain, jaise x aur y, aur hume in dono variables ki values find karni hoti hain.

Example:

2x + 3y = 13

3x − 2y = 4

Yahan hamara goal hai x aur y ki values find karna.

Pair of Linear Equations ko solve karne ke liye Class 10 Maths me different methods use kiye ja sakte hain. Inme Elimination Method ek important method hai. Is method me hum kisi ek variable ko equations se eliminate kar dete hain aur phir remaining variable ki value find karte hain.

Elimination Method ko agar basic logic ke saath samjha jaye, to ye method board exam me lengthy nahi lagta. Sabse important baat hai ki coefficients, signs aur multiplication ko carefully handle kiya jaye.

Standard Form of Linear Equation in Two Variables

Ek linear equation in two variables ka standard form hota hai:

ax + by + c = 0

Jab do linear equations hoti hain, to unhe generally is form me likha ja sakta hai:

a₁x + b₁y + c₁ = 0

a₂x + b₂y + c₂ = 0

Yahan:

• x, y → Variables
• a, b, c → Constants

Class 10 Board Exam me equations kabhi standard form me directly di ja sakti hain aur kabhi word problem se equations form karni pad sakti hain.

Elimination Method Kya Hai?

Elimination Method ka basic idea bahut simple hai:

Make one variable's coefficient equal → Add/Subtract → Variable disappears

Yani hum equations ko suitable numbers se multiply karte hain taaki kisi ek variable ke coefficients equal ya opposite ho jayein.

Uske baad equations ko add ya subtract karte hain. Is process me ek variable eliminate ho jata hai aur sirf ek variable remaining rehta hai.

Us variable ki value find karne ke baad use original equation me substitute karke doosre variable ki value find karte hain.

Elimination Method – Step by Step

Step 1: Equations Clearly Likhiye

Example:

2x + 3y = 13 ...(1)

3x − 2y = 4 ...(2)

Ab hume decide karna hai ki kis variable ko eliminate karna easier hoga.

Yahan y ke coefficients hain:

3 and −2

Inka LCM hai:

LCM = 6

Isliye hum y ko eliminate karne ke liye dono equations ko suitable numbers se multiply karenge.

Step 2: Coefficients Equal Kijiye

Equation (1) ko 2 se multiply karein:

4x + 6y = 26 ...(3)

Equation (2) ko 3 se multiply karein:

9x − 6y = 12 ...(4)

Ab y ke coefficients hain:

+6y and −6y

Ye opposite coefficients hain, isliye ab hum equations ko add kar sakte hain.

Step 3: Equations Add Kijiye

Equations:

4x + 6y = 26

9x − 6y = 12

Adding:

4x + 9x + 6y − 6y = 26 + 12

13x = 38

Therefore:

x = 38/13

Step 4: Second Variable Find Kijiye

Ab x = 38/13 ko original equation me substitute karte hain:

2x + 3y = 13

2(38/13) + 3y = 13

76/13 + 3y = 169/13

3y = 93/13

Therefore:

y = 31/13

Hence final answer:

x = 38/13

y = 31/13

Elimination Flow

Elimination Method ko yaad rakhne ke liye ye simple flow follow karein:

Choose Variable

↓

Make Coefficients Equal

↓

Add / Subtract

↓

One Variable Eliminated

↓

Find First Variable

↓

Substitute

↓

Find Second Variable

↓

Verify

Case 1: Same Coefficients → Subtract

Agar kisi variable ke coefficients already same hain, to equations ko subtract karke us variable ko eliminate kiya ja sakta hai.

Example:

3x + 2y = 12

3x − 4y = 6

Yahan x ke coefficients dono equations me 3 hain.

Subtract:

(3x + 2y) − (3x − 4y)

Brackets carefully remove karein:

3x + 2y − 3x + 4y = 12 − 6

6y = 6

Therefore:

y = 1

Case 2: Opposite Coefficients → Add

Agar kisi variable ke coefficients same magnitude ke hain lekin signs opposite hain, to equations ko add karke variable eliminate kiya ja sakta hai.

Example:

2x + 5y = 19

3x − 5y = 6

Yahan y ke coefficients +5 aur −5 hain.

Isliye equations ko add karein:

2x + 5y + 3x − 5y = 19 + 6

5x = 25

Therefore:

x = 5

Case 3: Coefficients Equal Nahi Hain

Agar coefficients equal ya opposite nahi hain, to pehle equations ko suitable numbers se multiply karna hota hai.

Example:

2x + 3y = 11 ...(1)

4x + 5y = 19 ...(2)

x ke coefficients hain:

2 and 4

Inka LCM:

LCM = 4

Equation (1) ko 2 se multiply karein:

4x + 6y = 22

Equation (2) already hai:

4x + 5y = 19

Ab subtract karein:

(4x + 6y) − (4x + 5y) = 22 − 19

y = 3

Ab original equation me y = 3 substitute karein:

2x + 3(3) = 11

2x + 9 = 11

2x = 2

Therefore:

x = 1

Final answer:

x = 1, y = 3

Case 4: Negative Coefficients

Negative coefficients wale questions me signs ko carefully handle karna bahut important hai.

Example:

3x − 2y = 5

2x + 2y = 10

Y ke coefficients −2 aur +2 hain.

Isliye equations ko add kar sakte hain:

3x − 2y + 2x + 2y = 5 + 10

5x = 15

Therefore:

x = 3

Ab original equation me substitute karein:

2(3) + 2y = 10

6 + 2y = 10

2y = 4

y = 2

Important Sign Rule

Elimination Method me sabse common mistakes me se ek sign error hai.

Jab complete equation subtract karte hain, to bracket ke andar har term ka sign carefully handle karna hota hai.

Example:

(4x + 6y) − (4x + 2y)

Bracket remove karne par:

4x + 6y − 4x − 2y

Therefore:

4y

⚠️ Sirf first term ka sign change karna galat hai. Subtraction ke waqt complete second equation ke signs carefully change karein.

Which Variable Should We Eliminate?

Har question me x ko hi eliminate karna zaroori nahi hai. Aap x ya y me se kisi bhi variable ko eliminate kar sakte hain.

Smart choice ye hai ki us variable ko choose karein jiske coefficients:

• Already equal hon.
• Already opposite hon.
• Smaller LCM produce karein.
• Calculation ko comparatively simple banayein.

Example:

2x + 7y = 15

3x + 5y = 13

x ke liye:

LCM(2,3) = 6

y ke liye:

LCM(7,5) = 35

Isliye x ko eliminate karna better choice hai, kyunki 6 se calculation 35 ke comparison me easier hogi.

Verification Kaise Kare?

Answer find karne ke baad verification karna ek useful habit hai. Isse calculation mistake pakadne me help milti hai.

Suppose:

x = 1, y = 3

Equation 1:

2x + 3y = 11

Values put karein:

2(1) + 3(3) = 2 + 9 = 11 ✅

Equation 2:

4x + 5y = 19

4(1) + 5(3) = 4 + 15 = 19 ✅

Dono equations satisfy ho rahi hain, isliye:

x = 1, y = 3

Important Formula Box

Pair of Linear Equations

a₁x + b₁y + c₁ = 0

a₂x + b₂y + c₂ = 0

Elimination Rule

Equal coefficients → Add/Subtract → Eliminate variable

If coefficients are opposite → ADD

If coefficients are same → SUBTRACT

If coefficients are different → Multiply first

Special Cases of Pair of Linear Equations

Pair of Linear Equations me generally three types of outcomes ho sakte hain: Unique Solution, No Solution aur Infinitely Many Solutions.

Unique Solution

Agar dono equations ki lines exactly ek point par intersect karti hain, to pair ka unique solution hota hai.

Condition:

a₁/a₂ ≠ b₁/b₂

Is case me x aur y ki ek definite pair of values milti hai.

No Solution

Agar dono equations parallel lines represent karti hain, to unka koi common point nahi hota. Isliye pair ka no solution hota hai.

Condition:

a₁/a₂ = b₁/b₂ ≠ c₁/c₂

Infinitely Many Solutions

Agar dono equations same line represent karti hain, to line ke har point par dono equations satisfy hongi. Isliye infinitely many solutions hote hain.

Condition:

a₁/a₂ = b₁/b₂ = c₁/c₂

Common Mistakes

Board Exam me Elimination Method ke questions solve karte waqt students commonly ye mistakes karte hain:

• ❌ Equation ko multiply karte waqt constant term ko multiply karna bhool jana.
• ❌ Sirf variable ke coefficient ko multiply karna aur baaki terms ko unchanged chhod dena.
• ❌ Subtraction ke time sirf ek term ka sign change karna.
• ❌ Opposite coefficients hone ke baad subtraction kar dena jabki addition se variable directly eliminate ho sakta tha.
• ❌ Coefficients equal kiye bina elimination start kar dena.
• ❌ Large LCM wala variable unnecessarily choose karna.
• ❌ Arithmetic mistakes karna.
• ❌ Final answer ko original equations me verify na karna.

Board Exam Question

Solve by Elimination Method:

2x + 3y = 13

3x − 2y = 4

Step 1: First Equation Multiply Karein

Equation (1) ko 2 se multiply karein:

4x + 6y = 26

Step 2: Second Equation Multiply Karein

Equation (2) ko 3 se multiply karein:

9x − 6y = 12

Step 3: Add Karein

Ab dono equations ko add karein:

4x + 6y + 9x − 6y = 26 + 12

13x = 38

Therefore:

x = 38/13

Step 4: Substitute

Original equation me x ki value put karein:

2(38/13) + 3y = 13

76/13 + 3y = 169/13

3y = 93/13

y = 31/13

Answer

x = 38/13

y = 31/13

Board Question – Easy

Solve:

x + y = 7

x − y = 3

Yahan y ke coefficients opposite hain, isliye equations ko add karenge:

x + y + x − y = 7 + 3

2x = 10

x = 5

Ab first equation me substitute karein:

5 + y = 7

y = 2

Answer → x = 5, y = 2

Board Question – Medium

Solve:

3x + 4y = 10

2x − 4y = 0

Y ke coefficients +4 aur −4 hain.

Isliye add karein:

3x + 4y + 2x − 4y = 10 + 0

5x = 10

x = 2

Substitute:

3(2) + 4y = 10

6 + 4y = 10

4y = 4

y = 1

Answer → x = 2, y = 1

Competency Question

A notebook and a pen together cost ₹50.

Two notebooks and three pens cost ₹120.

Let:

Notebook = x

Pen = y

According to the question:

x + y = 50 ...(1)

2x + 3y = 120 ...(2)

Equation (1) ko 2 se multiply karein:

2x + 2y = 100 ...(3)

Ab equation (3) ko equation (2) se subtract karein:

(2x + 3y) − (2x + 2y) = 120 − 100

y = 20

Ab equation (1) me y = 20 put karein:

x + 20 = 50

x = 30

Answer

Notebook = ₹30

Pen = ₹20

Quick Trick

Elimination Method ke liye ye three rules yaad rakhein:

SAME → SUBTRACT

OPPOSITE → ADD

NOT SAME → MULTIPLY FIRST

Ye shortcut aapko calculation start karne se pehle correct operation decide karne me help karega.

5-Second Revision

1. Write equations

↓

2. Choose variable

↓

3. Equalize coefficients

↓

4. Add/Subtract

↓

5. Find one variable

↓

6. Substitute

↓

7. Verify

Revision Box

• ✔ Pair of equations → 2 equations, 2 variables
• ✔ Standard form → ax + by + c = 0
• ✔ Elimination → Make coefficients equal or opposite
• ✔ Same coefficients → Subtract
• ✔ Opposite coefficients → Add
• ✔ Different coefficients → Multiply first
• ✔ Substitute to find the second variable
• ✔ Always verify the final answer
• ✔ Special cases → Unique / No / Infinite Solutions

Frequently Asked Questions (FAQs)

Elimination Method kya hai?

Elimination Method ek method hai jisme kisi ek variable ke coefficients ko equal ya opposite banakar equations ko add ya subtract kiya jata hai, jisse ek variable eliminate ho jata hai.

Elimination Method me pehle kya karna chahiye?

Sabse pehle decide karein ki x ya y me se kis variable ko eliminate karna easier hoga. Phir us variable ke coefficients ko equal ya opposite banayein.

Same coefficients hone par kya karna chahiye?

Agar coefficients same hain, to generally equations ko subtract karke variable eliminate kiya jata hai.

Opposite coefficients hone par kya karna chahiye?

Agar coefficients equal aur opposite signs me hain, to equations ko add karke variable eliminate kiya ja sakta hai.

Elimination Method me LCM ka use kyu hota hai?

LCM ka use coefficients ko equal banane ke liye kiya jata hai, jisse kisi ek variable ko easily eliminate kiya ja sake.

Pair of Linear Equations ke special cases kya hain?

Pair of Linear Equations ke three main cases hote hain: Unique Solution, No Solution aur Infinitely Many Solutions.

Elimination Method me answer verify karna zaroori hai?

Verification compulsory nahi hota, lekin ye highly recommended hai. Original equations me values put karne se calculation ya sign mistake identify karna easy ho jata hai.

Download PDF Notes

Class 10 Maths revision ke liye Pair of Linear Equations – Elimination Method ke PDF notes ko yahan directly read kar sakte hain:

Final Thoughts

Pair of Linear Equations – Elimination Method Class 10 Maths ka ek important topic hai. Is method ka main purpose kisi ek variable ko eliminate karke doosre variable ki value find karna hai.

Is method ko solve karte waqt sabse pehle equations ko clearly likhiye aur decide kijiye ki x ya y me se kis variable ko eliminate karna easiest hoga. Agar coefficients already same hain to subtract karein, agar opposite hain to add karein, aur agar coefficients different hain to suitable multiplication karke coefficients ko equal banayein.

Board Exam me calculation ke saath presentation bhi important hoti hai. Isliye equations ko proper numbering ke saath likhna, multiplication clearly show karna, elimination step dikhana aur final values ko clearly mention karna useful rahega.

Special cases me Unique Solution, No Solution aur Infinitely Many Solutions ke conditions ko bhi revise karna zaroori hai. Ye conditions conceptual aur competency-based questions me useful ho sakti hain.

Elimination Method ka complete flow yaad rakhiye:

Choose → Equalize → Add/Subtract → Eliminate → Substitute → Verify

Agar aap is flow ko practice ke saath use karte hain, to Pair of Linear Equations ke elimination-based questions ko systematically solve karna kaafi easy ho sakta hai.

Class 10 Board Exam preparation ke liye signs, multiplication aur final verification par special attention dijiye.