UP Board Class 10 Maths Chapter 3 Linear Equations: Substitution and Elimination Method Solved Simple Hindi Mein

“Pair of Linear Equations in Two Variables” – yeh naam padh ke hi bahut log confused ho jaate hain। Phir “Substitution Method”, “Elimination Method”, “Consistent/Inconsistent” jaise terms aate hain aur lagta hai koi badi complex cheez hai।

Lekin honestly? UP Board Class 10 Maths Chapter 3 Linear Equations actually ek bahut logical chapter hai. Jaise ek detective do clues se ek answer dhundhta hai – exactly waisi hi process hai yahan.

Aapke paas do equations hain. Dono mein x aur y hain. In donon ko use karke x aur y ki value nikalni hai. Bas yahi hai chapter

Aaj is post mein sab kuch zero se explain karenge – concepts, dono methods, word problems, aur exam ke liye practical tips. Ek ek step clearly.

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UP Board Class 10 Maths Chapter 3 Linear Equations: Pehle Chapter Samjho

Ek cheez pehle clear karte hain – is chapter mein hum sirf do cheezein seekhenge.

Pehli cheez hai yeh pehchaanna ki given equations ke kitne solutions hain – ek, zero, ya infinite। Doosri cheez hai woh solutions nikaalana jab exist karti ho.

Itna hi hai poora chapter. Baaki sab isi ke around hai.

Chapter mein topics:

Pehla topic hai linear equation kya hoti hai aur uski graphical representation/ Doosra hai pair of equations – consistent, inconsistent, dependent kya hote hain. Teesra aur sabse important hai algebraic methods – substitution aur elimination. Chautha hai word problems, jo real-life situations se equations banaakar solve karna.

Exam mein in sab se milke 15 se 18 marks aate hain UP Board mein.

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Linear Equation Kya Hai: UP Board Class 10 Maths Chapter 3 Ki Shuruat

Ek variable wali linear equation toh class 8-9 mein dekhi thi.

Jaise: 2x plus 5 equals 11। Isme sirf x hai. Ek step mein solve ho jaata hai.

Ab do variables wali linear equation aati hai.

Jaise: 2x plus 3y equals 12. Yahan x aur y dono unknown hain.

Ek equation se yeh solve nahi hoga. Iske liye dono equations milke kaam aati hain.

General form kya hoti hai:

a1x plus b1y plus c1 equals 0 … Equation 1
a2x plus b2y plus c2 equals 0 … Equation 2

Yahan a1, b1, c1 aur a2, b2, c2 real numbers hain aur a1 b2 ki value a2 b1 se equal nahi hoti.

Graphical: Har linear equation ek straight line represent karti hai. Do equations matlab do straight lines. Woh lines ya to ek point par milti hain, ya parallel chalti hain, ya ek doosre ke upar completely overlap karti hain.

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UP Board Class 10 Maths Chapter 3 Linear Equations: Consistent Inconsistent Kya Hota Hai

Yeh ek theory wala part hai jo exam mein 2 se 3 marks ka hota hai। Samajh aaya toh easy marks hain।

Case 1: Ek Solution (Intersecting Lines)

Dono lines ek point par milti hain. Is point pe x aur y ki jo value hai, wahi solution hai. Ek unique solution hota hai.

Isko kehte hain consistent pair of equations.

Case 2: Koi Solution Nahi (Parallel Lines)

Dono lines kabhi nahi milti। Koi solution nahi hota.

Isko kehte hain inconsistent pair of equations.

Case 3: Infinite Solutions (Coincident Lines)

Dono lines actually ek hi line hain – ek doosre ke upar overlap karti hain. Har point solution hai। Infinite solutions hain.

Isko kehte hain consistent and dependent pair of equations.

Ratio Test: UP Board Class 10 Maths Chapter 3 Linear Equations Mein Conditions

Bina graph banaye equations dekh ke solution type pata kar sakte ho। Yeh ratio test hai.

Equations ki general form lo:
a1x plus b1y plus c1 equals 0
a2x plus b2y plus c2 equals 0

Condition 1: a1/a2 not equal to b1/b2
Iska matlab: Unique solution. Lines intersect karti hain.

Condition 2: a1/a2 equals b1/b2 but not equal to c1/c2
Iska matlab: No solution. Lines parallel hain।

Condition 3: a1/a2 equals b1/b2 equals c1/c2
Iska matlab: Infinite solutions. Lines coincide karti hain.

Quick Example:

Equations: 2x plus 3y equals 9 aur 4x plus 6y equals 18

a1/a2 = 2/4 = 1/2
b1/b2 = 3/6 = 1/2
c1/c2 = 9/18 = 1/2

Teeno ratios equal hain! Matlab infinite solutions – coincident lines.

Doosra example:

Equations: x plus 2y equals 3 aur 2x plus 4y equals 9

a1/a2 = 1/2
b1/b2 = 2/4 = 1/2
c1/c2 = 3/9 = 1/3

a1/a2 equals b1/b2 but not equal to c1/c2. Matlab no solution – parallel lines.

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Substitution Method: UP Board Class 10 Maths Chapter 3 Linear Equations

Basic idea: Ek equation se ek variable ki value express karo doosre ke terms mein, phir yeh value doosri equation mein substitute karo. Single variable equation mil jaayegi, woh solve ho jaayegi.

Step-by-step process:

Step 1: Ek equation choose karo jisme ek variable ka coefficient 1 ya simple number ho.

Step 2: Usi variable ki value doosre variable ke terms mein nikalo.

Step 3: Yeh expression doosri equation mein substitute karo.

Step 4: Ab single variable wali equation solve karo.

Step 5: Jo value mili woh pehli equation mein daal ke doosra variable nikalo.

Step 6: Verification karo – dono equations mein values daal ke check karo.

UP Board Class 10 Maths Chapter 3 Linear Equations: Substitution Method Solved Example 1

Question: Solve: x plus y equals 14 aur x minus y equals 4

Solution:

x plus y equals 14 … (1)
x minus y equals 4 … (2)

Step 1: Equation (1) se x express karo.
x = 14 minus y … (3)

Step 2: Yeh value equation (2) mein substitute karo.
(14 minus y) minus y = 4
14 minus 2y = 4
minus 2y = 4 minus 14
minus 2y = minus 10
y = 5

Step 3: y = 5 ko equation (3) mein daal do.
x = 14 minus 5
x = 9

Answer: x = 9, y = 5

Verification: x plus y = 9 plus 5 = 14. Correct! x minus y = 9 minus 5 = 4. Correct!

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UP Board Class 10 Maths Chapter 3 Linear Equations: Substitution Method Solved Example 2

Question: Solve: 3x plus 4y equals 10 aur 2x minus 2y equals 2

Solution:

3x plus 4y equals 10 … (1)
2x minus 2y equals 2 … (2)

Step 1: Equation (2) ko simplify karo.
2x minus 2y = 2
x minus y = 1 (divide by 2)
x = 1 plus y … (3)

Step 2: Equation (1) mein substitute karo।
3(1 plus y) plus 4y = 10
3 plus 3y plus 4y = 10
3 plus 7y = 10
7y = 7
y = 1

Step 3: y = 1 ko equation (3) mein
x = 1 plus 1 = 2

Answer: x = 2, y = 1

Verification: 3(2) plus 4(1) = 6 plus 4 = 10. Correct! 2(2) minus 2(1) = 4 minus 2 = 2. Correct!

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UP Board Class 10 Maths Chapter 3 Linear Equations: Substitution Method Solved Example 3 (Fraction Wala)

Question: Solve: x/2 plus 2y/3 equals minus 1 aur x minus y/3 equals 3

Solution:

x/2 plus 2y/3 = minus 1 … (1)
x minus y/3 = 3 … (2)

Step 1: Equation (2) mein simplest variable x hai.
x = 3 plus y/3 … (3)

Step 2: Equation (1) mein x substitute karo.
(3 plus y/3)/2 plus 2y/3 = minus 1

Equation (1) ko pehle simplify karte hain – LCM of 2 and 3 is 6.
3x plus 4y = minus 6 … (1) multiply karke

Equation (2) multiply by 3:
3x minus y = 9 … (2)

Ab simpler ho gaya.

Step 3: Equation (1) revised se x nikalo.
3x = -6 – 4y
x = (- 6 – 4y)/3 … (3)

Step 4: Equation (2) revised mein dalo.
3 times (minus 6 minus 4y)/3 minus y = 9
(minus 6 minus 4y) minus y = 9
minus 6 minus 5y = 9
minus 5y = 15
y = minus 3

Step 5: y = -3 equation (3) mein.
x = (- 6 – 4 x -3) / 3
x = (- 6 + 12) / 3
x = 6/3
x = 2

Answer: x = 2, y = – 3

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Elimination Method: UP Board Class 10 Maths Chapter 3 Linear Equations

Doosra method aur bahut baar faster bhi lagta hai, especially jab coefficients convenient hon।

Basic idea: Dono equations ko aise multiply karo ki ek variable ka coefficient dono mein equal ho jaaye. Phir add ya subtract karo – woh variable “eliminate” ho jaayega. Ek variable ki equation mil jaayegi.

Step-by-step process:

Step 1: Decide karo kaunsa variable eliminate karna hai.

Step 2: Dono equations ko suitable numbers se multiply karo jisse us variable ke coefficients equal ho jaayein.

Step 3: Agar coefficients same sign ke hain toh subtract karo. Opposite sign ke hain toh add karo.

Step 4: Bacchi equation mein se remaining variable nikalo.

Step 5: Woh value original equation mein daal ke doosra variable nikalo.

Step 6: Verify karo.

UP Board Class 10 Maths Chapter 3 Linear Equations: Elimination Method Solved Example 4

Question: Solve: 3x + 4y = 10 or 2x – 2y = 2

Solution: (Same question, different method – compare karo!)

3x + 4y = 10 … (1)
2x – 2y = 2 … (2)

Step 1: y ko eliminate karte hain.

Equation (2) ko multiply by 2:
4x – 4y = 4 … (3)

Step 2: Equation (1) + Equation (3):
3x + 4y = 10
4x – 4y = 4

7x = 14
x = 2

Step 3: x = 2 equation (1) mein dalo
3(2) – 4y = 10
6 – 4y = 10
4y = 4
y = 1

Answer: x = 2, y = 1

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UP Board Class 10 Maths Chapter 3 Linear Equations: Elimination Method Solved Example 5 (Bigger Numbers)

Question: Solve: 8x + 5y = 9 aur 3x + 2y = 4

Solution:

8x + 5y = 9 … (1)
3x + 2y = 4 … (2)

Step 1: y ko eliminate karte hain.

Equation (1) multiply by 2:
16x + 10y = 18 … (3)

Equation (2) multiply by 5:
15x + 10y = 20 … (4)

Step 2: Equation (3) minus Equation (4):
16x + 10y = 18
– 15x – 10y = – 20

x = – 2

Step 3: x = – 2 equation (2) mein dalo.
3(- 2) + 2y = 4
– 6 + 2y = 4
2y = 10
y = 5

Answer: x = -2, y = 5

Verification:
Equation 1: 8(- 2) + 5(5) = – 16 + 25 = 9. Correct!
Equation 2: 3(- 2) + 2(5) = – 6 + 10 = 4. Correct!

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UP Board Class 10 Maths Chapter 3 Linear Equations: Elimination Method Solved Example 6

Question: Solve: 5x – 4y = – 8 aur x + 2y = 7

Solution:

5x – 4y = – 8 … (1)
x + 2y = 7 … (2)

Step 1: x ko eliminate karte hain।

Equation (2) multiply by 5:
5x + 10y = 35 … (3)

Step 2: Equation (3) minus Equation (1):
5x + 10y = 35
– 5x + 4y = 8 (signs reverse)

14y = 43

Hmm, yeh thoda ungainly aa raha hai. y ko eliminate karna better rahega.

Equation (2) multiply by 2:
2x + 4y = 14 … (3)

Equation (1) plus Equation (3):
5x – 4y = – 8
2x + 4y = 14

7x = 6
x = 6/7

x = 6/7 equation (2) mein:
6/7 + 2y = 7
2y = 7 – 6/7
2y = 49/7 – 6/7
2y = 43/7
y = 43/14

Answer: x = 6/7, y = 43/14

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UP Board Class 10 Maths Chapter 3 Linear Equations: Word Problems Kaise Solve Karein

Yeh section exam mein important hai aur bahut students yahan marks gawa dete hain.

Word problem mein real-life situation ko equations mein convert karna hota hai.

Strategy:

1. Do unknown quantities identify karo. Unhe x aur y name do.

2. Problem mein diye conditions se do equations banao.

3. Equations substitution ya elimination se solve karo.

4. Answer units ke saath likho.

5. Verify karo ki answer problem ke context mein sense banata hai.

UP Board Class 10 Linear Equations: Word Problem Solved Example 7

Question: Ek class mein total 40 students hain. Girls ki sankhya boys ki sankhya se 4 zyada hai। Kitne boys hain aur kitni girls?

Solution:

Maano boys ki sankhya = x
Maano girls ki sankhya = y

Condition 1: Total 40 students
x + y = 40 … (1)

Condition 2: Girls, boys se 4 zyada hain
y = x + 4 … (2)

Substitution method: (2) ko (1) mein dalo
x + (x + 4) = 40
2x + 4 = 40
2x = 36
x = 18

y = 18 + 4 = 22

Answer: Boys = 18, Girls = 22

Verification: 18 plus 22 = 40. Correct! 22 minus 18 = 4. Correct!

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UP Board Class 10 Linear Equations: Word Problem Solved Example 8 (Age Problem)

Question: Abhi se 5 saal baad, Rahul ki age Sita ki age se 3 guna hogi. Abhi se 5 saal pehle, Rahul ki age Sita ki age se 7 guna thi. Unki current age kya hai?

Solution:

Maano abhi Rahul ki age = x years
Maano abhi Sita ki age = y years

Condition 1: 5 saal baad Rahul = 3 times Sita
(x + 5) = 3(y + 5)
x + 5 = 3y + 15
x = 3y + 10
x – 3y = 10 … (1)

Condition 2: 5 saal pehle Rahul = 7 times Sita
(x – 5) = 7(y – 5)
x – 5 = 7y – 35
x = 7y – 30
x – 7y = – 30 … (2)

Elimination: (1) – (2):
(x – 3y) – (x – 7y) = 10 – (- 30)
x – 3y – x + 7y = 40
4y = 40
y = 10

x = 3(10) + 10 = 40

Answer: Rahul ki age = 40 years, Sita ki age = 10 years

Verification:
5 years baad: Rahul = 45, Sita = 15. 45 = 3 times 15. Correct!
5 years pehle: Rahul = 35, Sita = 5. 35 = 7 times 5. Correct!

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UP Board Class 10 Maths Chapter 3 Linear Equations: Speed Distance Problem Example 9

Question: Ek boat still water mein 8 km/hr ki speed se chalti hai. Stream ke saath (downstream) 45 km aur stream ke against (upstream) 27 km jaane mein same time lagta hai. Stream ki speed kya hai?

Solution:

Maano stream ki speed = y km/hr

Downstream speed = 8 + y
Upstream speed = 8 – y

Time downstream = 45/(8 + y)
Time upstream = 27/(8 – y)

Time equal hai:
45/(8 + y) = 27/(8 – y)

Cross-multiply:
45(8 – y) = 27(8 + y)
360 – 45y = 216 + 27y
360 – 216 = 27y + 45y
144 = 72y
y = 2

Answer: Stream ki speed = 2 km/hr

Yeh technically ek equation wala problem tha, lekin real life problems mein aise combination aate hain.

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UP Board Class 10 Maths Chapter 3 Linear Equations: Exam Ke Liye Important Tips

Kuch cheezein jo actually marks mein difference karti hain।

Tip 1: Method ka selection

Agar kisi ek equation mein ek variable ka coefficient 1 ya simple number hai – substitution easier hogi. Agar dono equations mein coefficients bade numbers hain – elimination better hai.

Tip 2: Equation number clearly likho

Har equation ko number do – Eq(1), Eq(2). Phir jab operations karo – “Multiplying Eq(1) by 3”, “Subtracting Eq(2) from Eq(3)” – yeh clearly likho. Examiner ko pata chalta hai aap kya kar rahe ho.

Tip 3: Verification mat bhoolna

Bahut log answer nikaal ke ruk jaate hain. Verification se 1 mark milta hai aur yeh confirm bhi karta hai ki answer sahi hai। 30 seconds lagte hain.

Tip 4: Word problems mein variables define karo

“Let x be the number of boys” ya “Let x = Rahul’s present age” – yeh line likho. Bina definition ke equation likhna partial marks deta hai.

Tip 5: Signs par dhyan do

Linear equations mein sabse common mistake sign error hai. Especially jab negative numbers ko multiply ya subtract karte ho. Ek ek step carefully karo.

Tip 6: Ratio test wala theory question

Agar exam mein puchha jaaye ki equations ka koi solution hai ya nahi, ya kitne solutions hain – ratio test lagao. a1/a2, b1/b2, c1/c2 nikalo aur compare karo. Answer clearly state karo.

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Practice Questions: UP Board Class 10 Maths Chapter 3 Linear Equations

Khud solve karo. Answers neeche hain.

Question 1: Solve by substitution: 2x + y = 5 aur 3x + 2y = 8

Question 2: Solve by elimination: 5a – 8b + 1 = 0 aur 3a – 24/5 + 4b/5 …

Simple version: 5x – 3y = 11 aur 3x + 5y = 7

Question 3: Kya in equations ka koi solution hai? Batao kyun.
2x + 3y = 8 aur 4x + 6y = 7

Question 4: Word problem: Ek number doosre se 27 zyada hai. Dono ka sum 91 hai. Numbers nikalo.

Question 5: Ek shop mein 5 pens aur 3 notebooks ki cost Rs 27 hai. 2 pens aur 4 notebooks ki cost Rs 22 hai. Ek pen aur ek notebook ki individual cost nikalo.

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Answers:

Question 1 ka answer:
y = 5 + 2x (from eq 1)
3x + 2(5 – 2x) = 8
3x + 10 – 4x = 8
– x = – 2
x = 2, y = 1

Question 2 ka answer:
5x – 3y = 11 … (1) multiply by 5: 25x – 15y = 55
3x + 5y = 7 … (2) multiply by 3: 9x + 15y = 21
Add: 34x = 76, x = 76/34 = 38/17

Hmm, simplify karein. Let me redo:
Eq(1) times 5: 25x – 15y = 55
Eq(2) times 3: 9x + 15y = 21
Add: 34x = 76
x = 38/17

x = 38/17 in eq (2): 3(38/17) + 5y = 7
114/17 + 5y = 7
5y = 7 – 114/17 = 119/17 – 114/17 = 5/17
y = 1/17

Question 3 ka answer:
a1/a2 = 2/4 = 1/2
b1/b2 = 3/6 = 1/2
c1/c2 = 8/7 (different!)
a1/a2 equals b1/b2 but not c1/c2 – Parallel lines, No solution।

Question 4 ka answer:
Let the numbers be x and y, where x is greater.
x – y = 27 … (1)
x + y = 91 … (2)
Add: 2x = 118, x = 59, y = 32
Answer: 59 aur 32

Question 5 ka answer:
Let pen = p, notebook = n
5p + 3n = 27 … (1)
2p + 4n = 22 … (2), simplified: p + 2n = 11 … (3)
From (3): p = 11 – 2n
In (1): 5(11 – 2n) + 3n = 27
55 – 10n + 3n = 27
– 7n = – 28
n = 4, p = 11 – 8 = 3
Answer: Pen = Rs 3, Notebook = Rs 4

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FAQ: UP Board Class 10 Maths Chapter 3 Linear Equations

1. UP Board Class 10 Maths Chapter 3 Linear Equations mein substitution ya elimination – kaunsa method better hai?

Dono equally valid hain। Agar ek variable ka coefficient 1 hai, substitution fast hoti hai. Agar dono equations mein bade coefficients hain, elimination cleaner hoti hai. Practice se khud pata chalega kab kaunsa use karna hai.

2. Consistent aur inconsistent equations mein kya fark hai?

Consistent equations ki koi na koi solution hoti hai – chahe ek ho ya infinite। Inconsistent equations ki koi solution nahi hoti। Ratio test se pata kar sakte ho – agar a1/a2 equals b1/b2 but c1/c2 alag hai – inconsistent hai।

3. Verification karna zaruri hai exam mein?

Haan, bilkul. Verification se confirm hota hai ki answer sahi hai. Aur bahut boards mein verification ke bhi marks hote hain. Verification mein 1 minute extra kharcha karo.

4. Word problems mein equations kaise banate hain – samajh nahi aata?

Ek formula hai. Pehle do unknown values identify karo. Unhe variable name do – x aur y. Phir problem mein diye do conditions se do equations banao. Phir solve karo. Practice se yeh skill develop hoti hai.

5. Kya equations mein fraction aa sakte hain answer mein?

Haan, Answers hamesha clean integers nahi hote. Fraction answer bhi valid hai. Bas correctly solve karo aur verify karo.

6. Infinite solutions wale case mein kya likhein exam mein?

Ratios check karo – a1/a2 equals b1/b2 equals c1/c2. Phir likho “The given pair of linear equations has infinitely many solutions” aur reason batao ki ratios equal hain.

7. UP Board Class 10 Maths Chapter 3 se kaunse types ke questions most common hain?

Substitution se solve karo (3 marks), elimination se solve karo (3 marks), solution type identify karo ratio se (2 marks), aur word problems (5 marks) – yeh sab most common hain। Inhe pakka karo

8. Elimination mein add karein ya subtract – confuse hota hai?

Rule simple hai: agar eliminate karne wale variable ke coefficients same sign ke hain (dono positive ya dono negative) – subtract karo. Agar opposite signs ke hain (ek positive, ek negative) – add karo.

9. Equations solve karne ke baad answer kaise present karein?

Clearly likho: x = [value] aur y = [value]। Word problems mein context add karo jaise “Number of boys = 18” ya “Speed of stream = 2 km/hr”. Verification separately likho.

10. Chapter 3 mein graphical method bhi hai – woh exam mein aata hai kya?

Graphical method ka basic concept – lines intersect, parallel ya coincide – samajhna chahiye. Actual graph banane ke questions sometimes aate hain, lekin algebraic methods (substitution, elimination) zyada marks ke hote hain. Algebraic methods ko priority do.

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Conclusion

UP Board Class 10 Maths Chapter 3 Linear Equations ek aisa chapter hai jo ek hi baar achi tarah padh lene layak hai

Do equations hain। Do unknown hain. Substitution se ya elimination se solve karo. Word problems mein equations banao, phir same methods lagao.

Ek practical cheez: Dono methods ki practice karo ek hi question par. Jaise upar dekha – same question substitution se bhi aur elimination se bhi solve kiya। Dono baar same answer aaya। Yeh confidence deta hai।

Abhi jo best kaam kar sakte ho woh hai: Practice questions nikaalkar khud solve karo। Galat jawab aaye toh dekho kahan galti hui – woh specific step ek baar phir padho।

Chapter 1 ke baad yeh Chapter 3 bhi clear ho jaaye toh exam mein accha feel hoga।

Important: NCERT ki official website ncert.nic.in par Class 10 Maths ki free PDF available hai. Chapter 3 wahan se download karo. UP Board exam mein NCERT level ke questions hi primarily aate hain aur exercises ki practice sabse effective preparation hai.

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