Class 10 Maths · Chapter 3
Short answer:
Is chapter me do exercises hain (Ex 3.1 aur Ex 3.2 — current rationalised NCERT me cross-multiplication method aur "equations reducible to a pair of linear equations" section trim ho chuke hain). Coverage: linear equations ko situations se banana, unhe graph pe represent karna, phir substitution method aur elimination method se algebraically solve karna — including age, cost, digit aur speed-distance word problems.
Do unknowns, do equations — aur unhe ek saath solve karne ka tareeka yahi chapter sikhata hai. Class 9 me ek linear equation me infinite solutions hote the (ek line); yahan do equations ek saath honge, aur unka common solution (agar exist kare) ek point hota hai jahan dono lines milti hain. Real life me ye tab aata hai jab do quantities (jaise do items ke daam, ya do logon ki age) ek doosre se related ho aur dono ke baare me alag-alag info di ho.
Chapter 3 Summary — 5 Minute Revision
1. Pair of Linear Equations — Basics
Do linear equations in two variables x aur y jab ek saath likhi jaayein, unhe "pair of linear equations" kehte hain:
| General form | a1x + b1y + c1 = 0 aur a2x + b2y + c2 = 0 (jahan a1,b1,a2,b2 dono zero nahi ho sakte) |
|---|
↔ Table ko side me swipe karein
Solution woh (x, y) pair hai jo dono equations ko satisfy kare — graph pe woh point jahan dono lines intersect (ya coincide) karti hain.
2. Graphical Method
Dono equations ko line ke form me plot karo (har equation ke liye kam se kam do points nikaalo, table banao). Teen possibilities hoti hain:
| Lines | Solutions | Nature |
|---|---|---|
| Intersecting | Exactly one (unique) | Consistent |
| Coincident (same line) | Infinitely many | Consistent (dependent) |
| Parallel | None | Inconsistent |
↔ Table ko side me swipe karein
3. Consistency Condition (bina graph banaye pata karna)
a1x + b1y + c1 = 0 aur a2x + b2y + c2 = 0 ke coefficients ka ratio compare karo:
| Condition | Graph | Solutions |
|---|---|---|
| a1/a2 ≠ b1/b2 | Intersecting lines | Unique solution — consistent |
| a1/a2 = b1/b2 = c1/c2 | Coincident lines | Infinitely many — consistent (dependent) |
| a1/a2 = b1/b2 ≠ c1/c2 | Parallel lines | No solution — inconsistent |
↔ Table ko side me swipe karein
4. Substitution Method
Ek equation se ek variable ko doosre ke terms me nikaalo, phir doosri equation me substitute kar do — ek variable ka equation ban jaata hai, jise solve karke wapas pehli equation me daalo.
5. Elimination Method
Elimination method me maksad ek variable ka coefficient barabar karke usse cancel karna hota hai — subtraction ya addition se. Dono equations ko aise constants se multiply karo ki ek variable ke coefficients equal (same sign to subtract, opposite sign to add) ho jaayein, phir add/subtract karke wo variable eliminate karo.
6. Word Problems ka approach
- Do unknown quantities ko clearly x aur y define karo (likh kar rakho — "let x = ...").
- Har condition ko ek-ek linear equation me convert karo.
- Substitution ya elimination se solve karo, phir dono original equations me check karo.
- Final answer ko question ke context me likho (sirf x, y nahi — "cost of bat = Rs 500" jaisa).
Poore Class 10 Maths ke handwritten colour notes
IITian & district toppers ke banaye short notes — revision-ready, diagram ke saath. Board se pehle poora syllabus 3 din me revise.
Exercise Questions — Solutions (Q1–Q16)
Ex 3.1, Q1. Aftab apni beti se kehta hai, "7 saal pehle main tumse 7 guna bada tha. Aur 3 saal baad main tumse sirf 3 guna bada hounga." Is situation ko algebraically aur graphically represent karo.
Aftab ki present age = x, beti ki present age = y le lo.
7 saal pehle: (x − 7) = 7(y − 7) ⇒ x − 7y + 42 = 0
3 saal baad: (x + 3) = 3(y + 3) ⇒ x − 3y − 6 = 0
Ye do equations hain. Graph pe dono lines banoge to woh ek point pe intersect karengi — wahi Aftab aur uski beti ki present ages hongi.
Ex 3.1, Q2. 2 kg apples aur 1 kg grapes ka daam Rs 160 hai. Ek mahine baad, 4 kg apples aur 2 kg grapes ka daam Rs 300 hai. Is situation ko algebraically aur geometrically represent karo.
Apples ka daam/kg = x, grapes ka daam/kg = y.
2x + y = 160
4x + 2y = 300 ⇒ 2x + y = 150
Dono equations me x aur y ke coefficients ka ratio same hai (2:1) par constant term alag hai (160 vs 150) — matlab a1/a2 = b1/b2 ≠ c1/c2. Ye parallel lines hain, koi solution nahi (inconsistent) — jo real life me bhi sahi hai kyunki daam badal gaya hai, dono conditions ek saath sahi nahi ho sakti.
Ex 3.2, Q1. 10 students ne ek maths quiz me part liya. Agar girls ki number boys se 4 zyada hai, to boys aur girls ki number nikaalo (graphically bhi represent karo).
Boys = x, girls = y.
x + y = 10
y = x + 4 ⇒ x − y = −4
Dono add karo: (x+y) + (x−y) = 10 + (−4) ⇒ 2x = 6 ⇒ x = 3
y = 10 − 3 = 7
Boys = 3, Girls = 7. Graph pe dono lines x = 3, y = 7 pe intersect karengi.
Ex 3.2, Q2. 5 pencils aur 7 pens ka total daam Rs 50 hai, jabki 7 pencils aur 5 pens ka total daam Rs 46 hai. Ek pencil aur ek pen ka daam graphically nikaalo.
Pencil ka daam = x, pen ka daam = y.
5x + 7y = 50 ...(i)
7x + 5y = 46 ...(ii)
(i)×5: 25x + 35y = 250
(ii)×7: 49x + 35y = 322
Subtract karo: 24x = 72 ⇒ x = 3
(i) me daalo: 5(3) + 7y = 50 ⇒ 7y = 35 ⇒ y = 5
Pencil = Rs 3, Pen = Rs 5.
Ex 3.2, Q3. Bina graph banaye batao ki ye pairs intersecting, parallel ya coincident lines hain: (a) 5x − 4y + 8 = 0 aur 7x + 6y − 9 = 0 (b) 9x + 3y + 12 = 0 aur 18x + 6y + 24 = 0 (c) 6x − 3y + 10 = 0 aur 2x − y + 9 = 0
a1/a2, b1/b2, c1/c2 compare karo:
(a) a1/a2 = 5/7, b1/b2 = −4/6 = −2/3. 5/7 ≠ −2/3 ⇒ intersecting (unique solution)
(b) a1/a2 = 9/18 = 1/2, b1/b2 = 3/6 = 1/2, c1/c2 = 12/24 = 1/2 ⇒ sab equal ⇒ coincident (infinite solutions)
(c) a1/a2 = 6/2 = 3, b1/b2 = −3/−1 = 3, c1/c2 = 10/9 ⇒ a1/a2 = b1/b2 ≠ c1/c2 ⇒ parallel (no solution)
Ex 3.3, Q1. Substitution method se solve karo: x + y = 14, x − y = 4.
x + y = 14 ...(i) ⇒ x = 14 − y
(ii) me daalo: (14 − y) − y = 4 ⇒ 14 − 2y = 4 ⇒ 2y = 10 ⇒ y = 5
x = 14 − 5 = 9
x = 9, y = 5
Ex 3.3, Q2. Substitution method se solve karo: s − t = 3, s/3 + t/2 = 6.
s − t = 3 ⇒ s = t + 3 ...(i)
Doosri equation ko 6 se multiply karo: 2s + 3t = 36 ...(ii)
(i) ko (ii) me daalo: 2(t+3) + 3t = 36 ⇒ 2t + 6 + 3t = 36 ⇒ 5t = 30 ⇒ t = 6
s = 6 + 3 = 9
s = 9, t = 6
Ex 3.3, Q3. 0.2x + 0.3y = 1.3 aur 0.4x + 0.5y = 2.3 ko solve karo.
(i)×2: 0.4x + 0.6y = 2.6
Isse (ii) subtract karo: (0.4x + 0.6y) − (0.4x + 0.5y) = 2.6 − 2.3 ⇒ 0.1y = 0.3 ⇒ y = 3
(i) me daalo: 0.2x + 0.3(3) = 1.3 ⇒ 0.2x = 0.4 ⇒ x = 2
x = 2, y = 3
Ex 3.3, Q4 (word problem). Do numbers ka difference 26 hai, aur ek number doosre ka 3 guna hai. Numbers nikaalo.
Bada number = x, chhota number = y.
x − y = 26 ...(i)
x = 3y ...(ii)
(ii) ko (i) me daalo: 3y − y = 26 ⇒ 2y = 26 ⇒ y = 13
x = 3(13) = 39
Numbers hain 39 aur 13.
Ex 3.3, Q5 (word problem). Do supplementary angles me se bada angle chhote se 18° zyada hai. Dono angles nikaalo.
Supplementary angles ka sum 180° hota hai. Bada angle = x, chhota = y.
x + y = 180 ...(i)
x − y = 18 ...(ii)
(i) + (ii): 2x = 198 ⇒ x = 99
y = 180 − 99 = 81
Angles hain 99° aur 81°.
Ex 3.4, Q1 (word problem). Cricket team ki coach 7 bats aur 6 balls Rs 3800 me kharidti hai. Baad me 3 bats aur 5 balls Rs 1750 me kharidti hai. Ek bat aur ek ball ka daam nikaalo (elimination method).
Bat ka daam = x, ball ka daam = y.
7x + 6y = 3800 ...(i)
3x + 5y = 1750 ...(ii)
(i)×5: 35x + 30y = 19000
(ii)×6: 18x + 30y = 10500
Subtract karo: 17x = 8500 ⇒ x = 500
(ii) me daalo: 3(500) + 5y = 1750 ⇒ 1500 + 5y = 1750 ⇒ 5y = 250 ⇒ y = 50
Bat = Rs 500, Ball = Rs 50.
Ex 3.4, Q2 (word problem). Ek fraction 2 add karne pe (numerator aur denominator dono me) 9/11 ban jaati hai, aur 3 add karne pe 5/6 ban jaati hai. Fraction nikaalo.
Fraction = x/y.
(x+2)/(y+2) = 9/11 ⇒ 11x + 22 = 9y + 18 ⇒ 11x − 9y = −4 ...(i)
(x+3)/(y+3) = 5/6 ⇒ 6x + 18 = 5y + 15 ⇒ 6x − 5y = −3 ...(ii)
(i)×5: 55x − 45y = −20
(ii)×9: 54x − 45y = −27
Subtract karo: x = 7
(i) me daalo: 11(7) − 9y = −4 ⇒ 77 − 9y = −4 ⇒ 9y = 81 ⇒ y = 9
Fraction = 7/9.
Ex 3.4, Q3 (word problem). Places A aur B ek highway pe 100 km door hain. Ek car A se aur doosri B se same time pe chalti hai. Same direction me chalein to 5 ghante me milti hain; opposite direction me (ek doosre ki taraf) chalein to 1 ghante me milti hain. Dono cars ki speed nikaalo.
A wali car ki speed = x km/h, B wali ki speed = y km/h (x > y).
Same direction: 5x − 5y = 100 ⇒ x − y = 20 ...(i)
Opposite direction: x + y = 100 ...(ii)
(i) + (ii): 2x = 120 ⇒ x = 60
y = 100 − 60 = 40
Speeds hain 60 km/h aur 40 km/h.
Ex 3.4, Q4 (word problem). Meena bank se Rs 2000 nikaalti hai, sirf Rs 50 aur Rs 100 ke notes me. Total 25 notes milte hain. Har type ke kitne notes mile?
Rs 50 ke notes = x, Rs 100 ke notes = y.
x + y = 25 ...(i)
50x + 100y = 2000 ⇒ x + 2y = 40 ...(ii)
(ii) − (i): y = 15
x = 25 − 15 = 10
Rs 50 ke 10 notes, Rs 100 ke 15 notes.
Ex 3.4, Q5 (word problem). Ek lending library ka fixed charge pehle 3 din ke liye hai, aur uske baad har extra din ka alag charge hai. Saritha ne 7 din ki book ke liye Rs 27 diye, Susy ne 5 din ki book ke liye Rs 21 diye. Fixed charge aur per-day extra charge nikaalo.
Fixed charge = Rs x, extra charge/din = Rs y.
7 din = 3 fixed + 4 extra din: x + 4y = 27 ...(i)
5 din = 3 fixed + 2 extra din: x + 2y = 21 ...(ii)
(i) − (ii): 2y = 6 ⇒ y = 3
(ii) me daalo: x + 2(3) = 21 ⇒ x = 15
Fixed charge = Rs 15, extra charge = Rs 3/din.
Ex 3.4, Q6 (word problem). Ek two-digit number aur uske digits ulta karke bane number ka sum 66 hai. Digits me 2 ka difference hai. Number nikaalo.
Tens digit = x, units digit = y. Number = 10x + y, reversed = 10y + x.
(10x + y) + (10y + x) = 66 ⇒ 11x + 11y = 66 ⇒ x + y = 6 ...(i)
Digits me difference 2 hai — do cases: x − y = 2 ...(ii)
(i) + (ii): 2x = 8 ⇒ x = 4, y = 2 ⇒ Number = 42
Agar y − x = 2 le to: x = 2, y = 4 ⇒ Number = 24
Number 42 ya 24 ho sakta hai (dono conditions satisfy karte hain — digits ka difference sirf magnitude me 2 diya tha, sign nahi).
Important Equations — Ek Nazar Me
| Method | Kab use karo | Key steps |
|---|---|---|
| Graphical | Jab specifically bola jaye "graphically solve karo", ya lines ki nature dikhani ho | Har equation ke liye kam se kam 2 points nikaalo → plot karo → intersection point = solution |
| Substitution | Jab ek equation me ek variable easily isolate ho jaaye (coefficient 1 ho) | Ek variable ko doosre ke terms me likho → doosri equation me daalo → ek-variable equation solve karo → wapas substitute karo |
| Elimination | Jab dono equations me coefficients aasani se match ho jaayein (LCM se multiply karke) | Ek variable ke coefficients equal karo (multiply karke) → same sign ho to subtract, opposite sign ho to add → ek variable milega → wapas daal kar doosra nikaalo |
↔ Table ko side me swipe karein
Consistency condition (a1x + b1y + c1 = 0, a2x + b2y + c2 = 0)
| Ratio comparison | Lines | Solutions | Type |
|---|---|---|---|
| a1/a2 ≠ b1/b2 | Intersecting | Unique (ek) | Consistent |
| a1/a2 = b1/b2 = c1/c2 | Coincident | Infinitely many | Consistent (dependent) |
| a1/a2 = b1/b2 ≠ c1/c2 | Parallel | None | Inconsistent |
↔ Table ko side me swipe karein
Common Mistakes — Yahan Marks Kat te Hain
- Elimination me sign error. Coefficients same sign ho to subtract karna hai, opposite sign ho to add — ye ulta kar dene se poora answer galat aa jaata hai. Multiply karte waqt bhi negative sign carry karna mat bhoolo.
- a1/a2, b1/b2, c1/c2 condition check kiye bina seedha solve karna shuru karna. Agar lines parallel hain to solution hai hi nahi — bina check kiye elimination karte rehne se time waste hota hai aur galat 'solution' bhi mil sakta hai (jaise 0 = 0 ya 0 = koi non-zero number aa jaana, jise interpret karna zaroori hai).
- Fractions/decimals clear karte waqt har term ko LCM se multiply na karna. Agar equation me x/3 + y/2 = 6 hai to LCM (yahan 6) se sirf kuch terms multiply karke baaki chhod dena bahut common mistake hai — har single term (constant sahit) multiply hona chahiye.
- Word problem me variable assignment mix-up. Boys/girls, apples/grapes, bat/ball jaise do quantities ko x aur y assign karte waqt confuse ho jaana — phir final answer me values swap ho jaati hain. Hamesha shuru me likh lo "let x = ...".
- Solution ko dono original equations me verify na karna. Ek equation se value nikaal kar seedha final answer maan lena risky hai — agar beech me koi arithmetic slip hui ho to wo pakdi nahi jaati. Dono equations me daal kar check zaroor karo.
- Graphical method me points bahut close-close le lena. Agar table of values me x ke liye sirf 0, 1 jaise close points liye jaayein, to line ka slope theek se plot nahi hota aur intersection point read karne me error aata hai — points thoda spread out (jaise 0, 2, 4) lo.
Board-Style Important Questions
- 1 mark: a1/a2, b1/b2, c1/c2 ke beech kya condition ho to pair of linear equations ka koi solution na ho?
- 2 marks: Substitution method se solve karo: 2x + 3y = 11 aur 2x − 4y = −24.
- 2 marks: Bina graph banaye batao ki x + 2y − 4 = 0 aur 2x + 4y − 12 = 0 se banne wali lines intersecting, parallel ya coincident hain.
- 3 marks: Do numbers ka sum 8 hai aur unka difference 2 hai. Elimination method se numbers nikaalo, aur solution ko verify karo.
- 5 marks: Ek word problem do jismein age ya cost involve ho, use pair of linear equations me convert karke elimination ya substitution method se solve karo, aur final answer ko context ke saath likho.
Quick Quiz — Score Check Karein
Q1. Class 10 Maths Chapter 3 (Pair of Linear Equations in Two Variables) me current rationalised NCERT ke hisaab se kitne exercises hain?
Q2. Agar do linear equations a1x+b1y+c1=0 aur a2x+b2y+c2=0 me a1/a2 = b1/b2 ≠ c1/c2 ho, to lines kaisi hongi?
Q3. Agar a1/a2 ≠ b1/b2 ho, to pair of linear equations ka kya nature hoga?
Q4. Aftab apni beti se kehta hai: '7 saal pehle main tumse 7 guna bada tha.' Agar Aftab ki present age x aur beti ki y ho, to is condition ka correct equation kaunsa hai?
Q5. 10 students ne maths quiz me part liya, jisme girls ki number boys se 4 zyada hai. Boys aur girls ki actual number kya nikli?
Q6. 5 pencils aur 7 pens ka total daam Rs 50 hai, jabki 7 pencils aur 5 pens ka Rs 46. Ek pencil aur ek pen ka daam kya hai?
Q7. Substitution method kab use karna best rehta hai (SPEC ke formulas table ke hisaab se)?
Q8. Do supplementary angles me se bada angle chhote se 18° zyada hai. Angles kya nikle?
Q9. Cricket coach 7 bats aur 6 balls Rs 3800 me, aur 3 bats aur 5 balls Rs 1750 me kharidti hai. Ek bat aur ek ball ka daam kya hai?
Q10. SPEC ke 'mistakes' section ke hisaab se, elimination method me sabse common galti kya hai?
Aksar Poochhe Jaane Wale Sawaal
Cross-multiplication method exam me use kar sakte hain kya?
Current rationalised NCERT chapter me cross-multiplication method nahi hai — sirf substitution aur elimination method syllabus me hain, wahi use karo.
Graphical method se solve karna zaroori hai kya?
Sirf tab jab question specifically 'graphically solve karo' bole ya lines ki nature (intersecting/parallel/coincident) dikhani ho. Warna algebraic method (substitution/elimination) fast aur accurate hota hai.
Kaise pata chalega ki equations ka unique solution hai, infinite solutions hain, ya koi solution nahi?
a1/a2 aur b1/b2 compare karo. Agar unequal hain to unique solution. Agar a1/a2 = b1/b2 = c1/c2 to infinite solutions. Agar a1/a2 = b1/b2 lekin c1/c2 alag hai to koi solution nahi.
Substitution ya elimination — kaun sa method use karein?
Agar kisi equation me ek variable ka coefficient already 1 hai (ya aasani se isolate ho sakta hai), substitution fast rehta hai. Agar dono equations me coefficients LCM se aasani se match ho jaate hain, elimination cleaner rehta hai.
Word problem me variable kaise choose karein?
Jo do quantities poochhi gayi hain unhi ko x aur y bana lo, aur shuru me clearly likh lo 'let x = ...', 'let y = ...' — isse confusion nahi hota final answer likhte waqt.
Agar dono equations same line represent karti hain (coincident) to kitne solutions hote hain?
Infinitely many solutions hote hain — dono equations actually same relationship represent kar rahi hain, isliye har point jo ek equation satisfy karta hai woh doosri bhi karega.
Class 10 Maths — Saare Chapters
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