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Pair of linear equations

Lesson 65 of 113

1. Ye topic hai kya?

Do unknown x, y. Har equation degree 1. Paper: solve, kitne solutions, k pe unique / infinite / none, do items ki keemat.

Yahan lock

ax+by+c=0, intercept se line, teen cases, a1/a2 vs b1/b2 vs c1/c2, substitution, elimination, cross-multiply, 1/x, two-equation word.

Yahan kya nahi

Parabola / zeros — Polynomials. (−b±√D)/2a — Quadratic equations. Distance, section — Coordinate. Pita–beta — Age. Boats — TSD.

2. Ek linear equation — poori line

ax + by + c = 0, a aur b dono saath zero nahi.
Infinitely many solutions — har point us line pe. Ek equation se (x,y) unique nahi.

2x + 3y − 6 = 0. y=0 → x=3 (x-intercept). x=0 → y=2 (y-intercept). (3,0) aur (0,2) jodo. (1.5, 1) bhi: 3+3−6=0.

b≠0: y = (−a/b)x − c/b. Slope = −a/b — sirf parallel check. Distance formula yahan nahi.
b=0: vertical line x = −c/a (jaise x−4=0).
x+y=5 ko “x=5” mat likhna. (0,5), (2,3), (5,0) sab. Unique tab doosri line kaate.
xy=6 ya x²+y=1 — is pair ke linear nahi.

3. Do lines — teen pictures

Har equation ek line. Solution = point jo dono pe ho.

Intersect · unique

Alag slope. Ek milta point. Consistent, independent.

Parallel · none

Same slope, alag intercept. Inconsistent.

Coincident · infinite

Doosri = pehli × number. Same line. Consistent, dependent.

Do lines. Polynomials ki parabola (x-axis cut) alag. Doosri line se milna ≠ x² ka zero.

Intercept sketch: 2x+3y=6 → (3,0) aur (0,2). Doosri equation ke intercept. Milti dikhein to unique lagti; algebraic se confirm.

4. Ratio test — graph ke baghair

Dono ko a1x + b1y + c1 = 0 aur a2x + b2y + c2 = 0. Constant left, sign ke saath.

Unique (intersect): a1/a2 ≠ b1/b2. c ki zaroorat nahi.
Infinite (same line): a1/a2 = b1/b2 = c1/c2
None (parallel): a1/a2 = b1/b2 ≠ c1/c2

Kyun? Slope = −a/b. a1/a2 = b1/b2 ⇒ slopes equal. Intercept tab same jab c bhi wahi ratio. Warna parallel alag.

Pair (=0 form)a1/a2b1/b2c1/c2Case
2x+y−7=0, x−y−2=02−17/2a ≠ b → unique
2x+3y−6=0, 4x+6y−12=01/21/21/2teen equal → infinite
2x+3y−6=0, 4x+6y−10=01/21/23/5a=b ≠ c → none
Diya 2x+3y=6 aur 4x+6y=12. Left lao: −6, −12. Ya dono ko same shape mein compare: 2/4=3/6=6/12. Ek =0 doosri =k mix mat karo — c ka sign toot-ta.
Unique ke liye sirf a1/a2 ≠ b1/b2. Infinite ke liye teen equal — sirf a=b se infinite galat, wo none ho sakta.
a2=0 ho to a1/a2 mat likho. Ek mein x hai doosri vertical/nahi — case separately.

k dhoondho — har baar teen ratios

k ek coeff mein. Unique / infinite / none alag values ho sakti. Kabhi koi case possible hi nahi.

Example · k · infinite possible

x + 2y − 4 = 0, 2x + ky − 8 = 0. Unique / infinite / none?

Solution

  1. Ratios

    a1/a2=1/2. b1/b2=2/k. c1/c2=(−4)/(−8)=1/2.

  2. Unique

    1/2 ≠ 2/k → k ≠ 4. (Cross: k·1 ≠ 4.)

  3. k=4

    b=2/4=1/2. Teen ratios 1/2 → infinite. c pehle se match, isliye none is pair pe aata hi nahi.

Answerk≠4 unique; k=4 infinite; none impossible
Example · k · none possible

2x + 3y − 7 = 0, 2x + ky − 14 = 0.

Solution

  1. Ratios

    a=2/2=1. b=3/k. c=−7/−14=1/2.

  2. Unique

    1 ≠ 3/k → k ≠ 3.

  3. k=3

    b=1, a=1, c=1/2 ≠ 1 → none. Infinite tab hota jab c bhi 1 hota (constant −7, −7) — yahan −14 hai, isliye infinite nahi.

Answerk≠3 unique; k=3 none; infinite impossible

5. Substitution

Ek equation se ek variable nikaalo, doosri mein rakh do. Tab use jab kisi ka coeff 1 ya −1 ho — divide kam.

Example · substitution

3x + 2y = 11, x − y = 3. (x,y)?

Solution

  1. Aasan wala

    Doosri: x = y+3.

  2. Pehli mein

    3(y+3)+2y=11 → 3y+9+2y=11 → 5y=2 → y=2/5.

  3. Wapas

    x=2/5+3=17/5. Check doosri: 17/5 − 2/5=15/5=3.

Answer(17/5, 2/5)

Pehli mein y nikaal ke bhi same. Jo chhota dikhe.

6. Elimination

Ek variable ke coeff barabar (multiply), phir jodo ya minus. LDC default jab dono coeffs bade.

Example · elimination

3x + 4y = 10, 2x − 3y = −1.

Solution

  1. x hatao? 6

    ×2 pehli, ×3 doosri: 6x+8y=20, 6x−9y=−3.

  2. Minus

    (6x+8y)−(6x−9y)=20−(−3) → 8y+9y=23 → 17y=23 → y=23/17.

  3. Better: y hatao 12

    Pehli×3, doosri×4: 9x+12y=30, 8x−12y=−4. Jodo: 17x=26 → x=26/17.

  4. y

    2(26/17)−3y=−1 → 52/17 − 3y=−1 → −3y=−1−52/17=−(17+52)/17=−69/17 → y=23/17.

Answer(26/17, 23/17)
Sign: minus karte waqt doosri ke minus plus. 20−(−3)=23. Jodna tab jab signs opposite hon (12y aur −12y).
Dono sides same number se multiply. Sirf left ×2, right bhoolna — common. Check: values wapas original mein.

7. Cross-multiplication

Dono already a1x+b1y+c1=0, a2x+b2y+c2=0. Formula ratne se pehle ek baar kyun.

x nikalne ke liye y hatao — NCERT formula yeh lock:

x / (b1c2 − b2c1) = y / (c1a2 − c2a1) = 1 / (a1b2 − a2b1)
Denominator 0: unique nahi (parallel ya coincident — ratio se decide).

Yaad: write coeffs row: a1 b1 c1 / a2 b2 c2. x ke neeche b,c skip a; y ke neeche c,a skip b; 1 ke neeche a,b skip c. Har pair “down-cross minus”.

Example · cross

2x + y − 7 = 0, x − y − 2 = 0.

Solution

  1. Coeffs

    a1=2, b1=1, c1=−7; a2=1, b2=−1, c2=−2.

  2. 1 wala

    a1b2−a2b1=2(−1)−1(1)=−2−1=−3 ≠ 0 → unique.

  3. x

    b1c2−b2c1=1(−2)−(−1)(−7)=−2−7=−9. x=−9/−3=3.

  4. y

    c1a2−c2a1=(−7)(1)−(−2)(2)=−7+4=−3. y=−3/−3=1.

  5. Check

    2·3+1=7, 3−1=2.

Answer(3, 1)
c = −7 ko +7 mat banao. Equation =0 form mein jo likha c wahi. 2x+y=7 ko pehle 2x+y−7=0.

8. Reducible — 1/x, 1/y

Dikhe quadratic, asl mein linear in 1/x. x≠0, y≠0.

2/x + 3/y = 13 → u=1/x, v=1/y → 2u+3v=13. Solve (u,v), phir x=1/u, y=1/v.
Example · 1/x

2/x + 3/y = 13, 5/x − 4/y = −2. x,y?

Solution

  1. u,v

    2u+3v=13, 5u−4v=−2.

  2. ×4, ×3

    8u+12v=52, 15u−12v=−6. Jod: 23u=46, u=2. x=1/2.

  3. v

    2(2)+3v=13 → 4+3v=13 → v=3. y=1/3.

Answerx=1/2, y=1/3
u=2 ka matlab x=2 nahi — x=1/u. 2/x ko 2x mat padhna.

9. Word — do cheezein, do equations

Unknown lock: do prices, do counts, two-digit 10p+q. Age (pita/beta) Age page. Speed boats TSD.

Har sentence ek linear. Do sentences → pair. Check: variables 2, equations 2, consistent.
Example · pens

7 pen + 5 pencil = ₹50. 5 pen + 7 pencil = ₹46. Ek pen, ek pencil?

Solution

  1. Lock

    p=pen, n=pencil. 7p+5n=50, 5p+7n=46.

  2. ×7, ×5

    49p+35n=350, 25p+35n=230. Minus: 24p=120, p=5.

  3. n

    7·5+5n=50 → 35+5n=50 → n=3. Check: 5·5+7·3=25+21=46.

Answerpen ₹5, pencil ₹3
Example · two-digit

Do-digit number. Digit-sum 9. Number, reverse se 9 zyada. Number?

Solution

  1. Lock

    Tens t, units u. Number=10t+u. Reverse=10u+t. t+u=9. 10t+u = (10u+t)+9.

  2. Simplify

    10t+u−10u−t=9 → 9t−9u=9 → t−u=1.

  3. With sum

    t+u=9, t−u=1. Jod: 2t=10, t=5, u=4. Number 54. Reverse 45, farq 9.

Answer54
Unit-digit tricks (last digit of 7^n) Number system page. Yahan 10t+u algebra.

10. Sawal — basic se pro

Q1 · Basic

Kaun sa linear pair hai (do variables)?

  1. x² + y = 3, x + y = 1
  2. 2x − y = 4, x + 3y = 7
  3. xy = 6, x + y = 5
  4. x + 2 = 0 sirf

Solution

  1. Degree 1, do eq

    B dono degree 1, two vars. A mein x². C mein xy. D ek equation — line, pair nahi.

AnswerB
Q2 · Basic

2x+3y−6=0, 4x+6y−10=0. Kitne solutions?

  1. Ek
  2. Infinite
  3. Zero
  4. Do

Solution

  1. Ratios

    a=2/4=1/2, b=3/6=1/2, c=6/10=3/5. 1/2=1/2 ≠ 3/5 → parallel, none.

AnswerC · Zero
Q3 · Basic

x+y=5, x−y=1. (x,y)?

  1. (3,2)
  2. (2,3)
  3. (5,1)
  4. (4,1)

Solution

  1. Jod

    2x=6, x=3. y=5−3=2. Check x−y=2. B ulta. D: 4−1=3≠1.

AnswerA · (3,2)
Q4 · Basic · intercept

3x+2y=12 ka y-intercept?

  1. 4
  2. 6
  3. 3
  4. 12

Solution

  1. x=0

    2y=12, y=6. X-intercept y=0 → x=4. D poora 12 nahi.

AnswerB · 6
Q5 · Medium · k

x+2y−4=0, 2x+ky−8=0. Infinite solutions ke liye k?

  1. 2
  2. 4
  3. 8
  4. 1

Solution

  1. Need a=b=c

    1/2 = 2/k = 4/8=1/2. Isliye 2/k=1/2 → k=4.

  2. A

    k=2: b=2/2=1 ≠ 1/2 → unique, infinite nahi.

AnswerB · 4
Q6 · Medium

2x+3y−7=0, 2x+ky−14=0. No solution ke liye k?

  1. 3
  2. 7
  3. 2
  4. 6

Solution

  1. a=1, c=1/2

    b=3/k. None: 1=3/k ≠ 1/2 → k=3, aur 1≠1/2 already. k=3.

  2. Infinite?

    c match nahi (1 vs 1/2), k=3 pe none, infinite is pair pe nahi.

AnswerA · 3
Q7 · Medium · sub

2x+3y=12, y=x−1. x?

  1. 3
  2. 4
  3. 5
  4. 2

Solution

  1. Plug

    2x+3(x−1)=12 → 2x+3x−3=12 → 5x=15 → x=3. y=2. Check 6+6=12.

AnswerA · 3
Q8 · Medium · cross

2x+y−7=0, x−y−2=0. Solution?

  1. (3,1)
  2. (1,3)
  3. (2,3)
  4. (7,2)

Solution

  1. Jod

    (2x+y)+(x−y)=7+2 → 3x=9, x=3. y=7−6=1. B (1,3): 2+3≠7.

AnswerA · (3,1)
Q9 · Pro · 1/x

1/x + 1/y = 5, 1/x − 1/y = 1. x?

  1. 1/3
  2. 3
  3. 1/2
  4. 2

Solution

  1. u=1/x, v=1/y

    u+v=5, u−v=1. Jod: 2u=6, u=3. x=1/3.

  2. Trap B

    u=3 ko x samajhna. v=2, y=1/2.

AnswerA · 1/3
Q10 · Pro · word

7 pen + 5 pencil = 50, 5 pen + 7 pencil = 46. Pen ki keemat?

  1. 3
  2. 5
  3. 7
  4. 4

Solution

  1. p,n

    7p+5n=50, 5p+7n=46. ×7 aur ×5: 49p+35n=350, 25p+35n=230. 24p=120, p=5.

AnswerB · 5

Practice

Pehle khud, phir neeche kholo.

12345 678910
P1

2x+3y−6=0, 4x+6y−12=0?

  1. Unique
  2. Infinite
  3. None
  4. Exactly two

Solution

  1. 1/2=1/2=1/2

    Doosri = 2× pehli. Same line, infinite. D: linear pair mein “exactly two” nahi hota.

AnswerB
P2

x+y=4 ka x-intercept?

  1. 4
  2. 0
  3. 2
  4. 1

Solution

  1. y=0

    x=4. Point (4,0). Y-intercept (0,4).

AnswerA · 4
P3

x+2y=7, 2x+4y=5. Solutions?

  1. Unique
  2. Infinite
  3. None
  4. (7,5)

Solution

  1. =0

    x+2y−7=0, 2x+4y−5=0. a=b=1/2, c=7/5 ≠ 1/2 → none. Doosri pehli ki 2× nahi (14≠5).

AnswerC
P4

3x−y=7, x=2. y?

  1. −1
  2. 1
  3. −7
  4. 13

Solution

  1. Plug

    6−y=7, y=−1. Vertical line x=2 kaatti pehli ko.

AnswerA · −1
P5

kx+2y−3=0, 2x+4y−6=0. Infinite ke liye k?

  1. 1
  2. 2
  3. 3
  4. 4

Solution

  1. b=c=1/2

    2/4=1/2, 3/6=1/2. a=k/2 = 1/2 → k=1. Unique: k≠1.

AnswerA · 1
P6

x + 2y = 8, 2x − y = 1. Elimination se y?

  1. 2
  2. 1
  3. 3
  4. 0

Solution

  1. Pehli ×2

    2x+4y=16. Doosri 2x−y=1. Minus: 5y=15, y=3.

  2. x

    x+6=8, x=2. Check 4−3=1.

AnswerC · 3
P7

Do-digit. Digit-sum 9, number reverse se 9 zyada. Number?

  1. 45
  2. 54
  3. 63
  4. 36

Solution

  1. t−u=1, t+u=9

    t=5, u=4 → 54. 54−45=9. A 45 to reverse se chhota. C: 63−36=27≠9.

AnswerB · 54
P8

2/x + 1/y = 5, 1/x − 1/y = 1. y?

  1. 1
  2. 1/2
  3. 2
  4. 1/3

Solution

  1. u,v

    2u+v=5, u−v=1. Jod: 3u=6, u=2. v=u−1=1. y=1/v=1.

AnswerA · 1
P9

a1/a2 ≠ b1/b2 ka matlab graph pe?

  1. Parallel
  2. Ek point pe milti
  3. Same line
  4. Parabola

Solution

  1. Unique

    Alag slope, intersect. D polynomials. C infinite. A none.

AnswerB
P10

3x+4y=10, 6x+8y=k. Unique solution ke liye k?

  1. Sirf 20
  2. Koi bhi k≠20
  3. Koi bhi k
  4. Impossible — unique kabhi nahi

Solution

  1. a=b=1/2 hamesha

    6/3=2, 8/4=2, slopes same. Unique tab a≠b — yahan kabhi nahi. k=20 coincident infinite; k≠20 none. D.

AnswerD

← Index · Polynomials

Agle topic: Quadratic equations.