Linear Equations

A+B=120/3=40. A-B=10.
2A=50 → A=25, B=15.

తెలుగు: వ్యతిరేక దిశలలో → A+B=40.
A-B=10 → A=25, B=15 km/h.

Together 4 days: 4(1/10+1/15)=4×5/30=2/3 done.
Remaining=1/3. B alone: (1/3)/(1/15)=5 days.

తెలుగు: 4 రోజులలో 2/3 పని.
మిగిలిన 1/3: B = 5 రోజులలో చేస్తాడు.

x = 10 - 3 = 7

తెలుగు: -x = 3-10 = -7 → x = 7.

C = 1/4-1/6 = 1/12. B = 1/4-1/8-wait.
A+B+C=1/4. A+B=1/6. B+C=1/8.
C=1/4-1/6=1/12. B=1/8-1/12=1/24. A=1/6-1/24=3/24=1/8.
A alone = 8 days.

తెలుగు: C=1/4-1/6=1/12. B=1/8-1/12=1/24.
A=1/6-1/24=1/8. A ఒంటరిగా = 8 రోజులు.

x=4, y=3: 12+12=24 ✓
Also x=0,y=6 or x=8,y=0.

తెలుగు: x=4, y=3 అయినపుడు: 3(4)+4(3)=12+12=24 ✓.
ఇది ఒక valid positive solution.

3k and 4k: 4k-3k=k=5 → k=5.
Numbers: 15 and 20.

తెలుగు: 4k-3k=k=5 → k=5.
సంఖ్యలు = 15 మరియు 20.

2x = 30 → x = 15

తెలుగు: 2x = 30 → x = 15.

From eq1: y=2x-1. Sub in eq2: x+2(2x-1)=8 → 5x=10 → x=2, y=3.
Point (2,3) is intersection.

తెలుగు: x=2, y=3. Graph లో ఇవి రెండు రేఖలు (2,3) వద్ద ఖండించుకుంటాయి.
Graphical solution = Point of intersection = (2,3).

x + 3 = 2x → x = 3

తెలుగు: x+3 = 2x → x=3.
Check: 3+3=6=2×3 ✓

Correct=x, Wrong=75-x.
4x-(75-x)=195 → 5x=270 → x=54

తెలుగు: 4x-(75-x)=195 → 5x=270 → x=54.
Check: 4(54)-21=216-21=195 ✓

3k+5k=48 → k=6 → 18 and 30

తెలుగు: 3k+5k=48 → 8k=48 → k=6.
సంఖ్యలు = 18 మరియు 30.

C=x, B=x+5, A=x+8.
3x+13=60 → x=47/3...

Adjust: A+B+C=62. C=x, B=x+5, A=x+8. 3x+13=62 → 3x=49. Not clean.
A+B+C=54: 3x+13=54 → x=41/3. Not clean.

A+B+C=63: 3x+13=63 → x=50/3. Not clean.
A+B+C=72: 3x+13=72 → x=59/3. Not.

Use A is 5 more than B, B is 3 more than C. Sum=60.
C=x, B=x+3, A=x+8. 3x+11=60 → 3x=49. Not.

Use sum=66: x=55/3. Not.
Use A 4 more than B, B 2 more than C, sum=57: 3x+6=57 → x=17. C=17,B=19,A=23. ✓

తెలుగు: C=17, B=19, A=23. Sum=59. Hmm.
C=17,B=19,A=21: 21=19+2, 19=17+2. Sum=57. C=17,B=19,A=21.

Let: A=B+4, B=C+2. Sum=57. C=x, B=x+2, A=x+6. 3x+8=57 → x=49/3. Not.

Sum=60, A=B+5, B=C+1: C=x,B=x+1,A=x+6. 3x+7=60 → 3x=53. Not.

Sum=60, A=B+6, B=C+2: 3x+8=60 → 3x=52. Not.

Sum=60, A=B+4, B=C+6: 3x+10=60 → 3x=50. Not.

Sum=63, A=B+3, B=C+5: 3x+8=63 → x=55/3. Not.

Hmm. Let me try: B=x, A=x+3, C=x-5. x+(x+3)+(x-5)=60 → 3x-2=60 → x=62/3. Not.

B=x, A=x+3, C=x-6: 3x-3=60 → x=21. A=24,B=21,C=15. Sum=60 ✓. A is 3 more than B. B is 6 more than C. ✓

తెలుగు: B=21, A=24, C=15. Sum=60 ✓.

1/t = 1/3+1/4-1/6 = 4/12+3/12-2/12 = 5/12
t = 12/5 = 2.4 hours

తెలుగు: 1/t = 1/3+1/4-1/6 = 5/12.
t = 12/5 = 2.4 గంటలు.

xy=18, 10x+y-(10y+x)=27 → 9x-9y=27 → x-y=3.
xy=18, x-y=3: x=6,y=3 → 63-36=27 ✓. Number=63.

తెలుగు: xy=18, x-y=3.
x=6, y=3 → సంఖ్య=63.
Check: 6×3=18 ✓, 63-36=27 ✓

-2x = 3 - 7 = -4 → x = 2

తెలుగు: 7 రెండు వైపుల నుండి తీసివేయాలి.
-2x = -4 → x = 2.
Check: 7-2(2) = 3 ✓

x + (x+1) = 25 → 2x+1=25 → x=12

తెలుగు: వరుస సంఖ్యలు x మరియు x+1.
2x+1=25 → x=12 (చిన్నది).

Eq2-Eq1: y+2z=8 ...(4)
Eq3-Eq2: y+3z=12 ...(5)
(5)-(4): z=4. y=0. x=2.

తెలుగు: Eq2-Eq1: y+2z=8.
Eq3-Eq2: y+3z=12.
తీసివేస్తే z=4, y=0, x=2.

x + x/4 = 25 → 5x/4 = 25 → x = 20

తెలుగు: x+x/4=25 → 5x/4=25 → x=20.
Check: 20+5=25 ✓

20x+30(25-x)=580 → 20x+750-30x=580 → -10x=-170 → x=17
17kg at ₹20, 8kg at ₹30.

తెలుగు: 20x+30(25-x)=580 → x=17.
₹20 కు 17 kg, ₹30 కు 8 kg.

60(t+2) = 90t → 60t+120 = 90t → 30t=120 → t=4 hours (after Train2 starts)

తెలుగు: 60(t+2)=90t → t=4 గంటలు (రైలు2 బయలుదేరిన తర్వాత).

x × 110/100 = 5500 → x = 5000

తెలుగు: x × 1.1 = 5500 → x = ₹5000.

From x+y=5: x=5-y. Substitute:
2(5-y)+3y=12 → 10-2y+3y=12 → y=2, x=3

తెలుగు: x=5-y ని మొదటి eq లో substitute చేయాలి.
y=2, x=3.
Check: 2(3)+3(2)=12 ✓, 3+2=5 ✓

Multiply eq2 by 2: 10x-2y=28. Add to eq1:
13x=44 → x=44/13...

Try: from eq2: y=5x-14. Sub in eq1:
3x+2(5x-14)=16 → 13x=44 → x=44/13. Not clean.

Use: 2x+3y=16 and 5x-y=14.
From eq2: y=5x-14. Sub: 2x+3(5x-14)=16 → 17x=58. Not clean.

Try 3x+2y=16, x+y=6: y=6-x, 3x+2(6-x)=16 → x=4, y=2. Clean!

Use: 3x+2y=16, x+y=6. x=4, y=2.

తెలుగు: x+y=6 → y=6-x.
3x+2(6-x)=16 → x=4, y=2.

2x = 11 - 3 = 8 → x = 4

తెలుగు: రెండు వైపుల నుండి 3 తీసివేయాలి.
2x = 8 → x = 4.
Check: 2(4)+3 = 11 ✓

x + 7 = 15 → x = 8

తెలుగు: x + 7 = 15 → x = 8.

v₁t₁ + v₂t₂ = D

This is the fundamental variable speed linear equation.
Example: 60×2+80×3=120+240=360km.

తెలుగు: v₁t₁+v₂t₂=D.
ఇది variable speed problem కు basic equation.

2x + 8 = 3x - 3 → x = 11

తెలుగు: 2x+8 = 3x-3.
11 = 3x-2x → x = 11.
Check: 2(15)=30, 3(10)=30 ✓

Son=x, Father=4x.
(4x+5) = 3(x+5) → 4x+5 = 3x+15 → x=10
Son=10, Father=40

తెలుగు: కొడుకు=x, తండ్రి=4x.
4x+5=3(x+5) → x=10.
కొడుకు 10, తండ్రి 40 సంవత్సరాలు.

x = 35/5 = 7

తెలుగు: రెండు వైపులా 5 తో భాగించాలి.
x = 7.

P × (0.8)³ = 576 → P × 0.512 = 576 → P = 1125

తెలుగు: P×0.8×0.8×0.8 = 576.
P×0.512 = 576 → P = ₹1125.

15/(8-v)+18/(8+v)=3.
15(8+v)+18(8-v)=3(64-v²)
120+15v+144-18v=192-3v²
3v²-3v-72=0... Hmm let me try.
264-3v=192-3v² → 3v²-3v+72=0 → v²-v+24=0. Discriminant negative.

Use: 10/(8-v)+15/(8+v)=3.
10(8+v)+15(8-v)=3(64-v²)
80+10v+120-15v=192-3v²
200-5v=192-3v² → 3v²-5v+8=0. Discriminant=25-96<0.Use: upstream 6km, downstream 12km, still water 6km/h, total 3hrs. 6/(6-v)+12/(6+v)=3. 6(6+v)+12(6-v)=3(36-v²) 36+6v+72-12v=108-3v² 108-6v=108-3v² 3v²=6v → v²=2v → v=2 km/h. Clean!తెలుగు: 6/(6-v)+12/(6+v)=3. v=2 km/h. Check: 6/4+12/8=1.5+1.5=3 ✓

CP × 120/100 = 480 → CP = 480×100/120 = 400

తెలుగు: CP × 1.2 = 480 → CP = 400.
Check: 400×120/100 = 480 ✓

1/4 + 1/6 = 3/12+2/12 = 5/12 per hour.
Time = 12/5 = 2.4 hours

తెలుగు: 1/t = 1/4+1/6 = 5/12.
t = 12/5 = 2.4 గంటలు.

Adding: 2x=10 → x=5. Then y=2.

తెలుగు: రెండు equations కూడితే 2x=10 → x=5.
తీసివేస్తే 2y=4 → y=2.

n+d=8. (n+1)/(d-1)=1 → n+1=d-1 → d-n=2.
n+d=8, d-n=2 → d=5, n=3. Fraction=3/5.

తెలుగు: n+d=8, d-n=2 → d=5, n=3.
Fraction = 3/5.

ax - cx = d - b → x(a-c) = d-b → x = (d-b)/(a-c)

తెలుగు: ax-cx = d-b.
x(a-c) = d-b → x = (d-b)/(a-c).

Subtracting: (p-q)x-(p-q)y=r-s → (p-q)(x-y)=r-s
x-y = (r-s)/(p-q)

తెలుగు: రెండు equations తీసివేస్తే (p-q)(x-y)=r-s.
x-y = (r-s)/(p-q).

If a/l=b/m but c/n≠b/m → parallel lines → no solution (inconsistent).
If a/l≠b/m → intersecting → unique solution (consistent).
If a/l=b/m=c/n → same line → infinite solutions.

తెలుగు: a/l≠b/m → రేఖలు ఖండించుకుంటాయి → unique solution.
a/l=b/m≠c/n → parallel → no solution.
Consistent = solution ఉన్న system.

x²+y² = (x+y)²-2xy = 100-42 = 58

తెలుగు: x²+y² = (x+y)²-2xy = 100-42 = 58.

Let total=1. Milk=3/4, Water=1/4.
Remove fraction f: Milk left=3(1-f)/4, Water left=(1-f)/4+f.
3(1-f)/4 / [(1-f)/4+f] = 2/3.
9(1-f) = 2(1-f+4f) = 2(1+3f)
9-9f = 2+6f → 7 = 15f → f = 7/15.

తెలుగు: 3(1-f)/4 / [(1-f)/4+f] = 2/3 → f=7/15.
తీసివేసిన fraction = 7/15.

x=3k, y=4k. (3k+8)/(4k+8)=5/6.
6(3k+8)=5(4k+8) → 18k+48=20k+40 → 2k=8 → k=4.
x=12, y=16.

తెలుగు: 3k+8)/(4k+8)=5/6 → k=4.
x=12, y=16.
Check: 12/16=3/4 ✓, 20/24=5/6 ✓

Tens=2y, Units=y. Number=20y+y=21y.
Reversed=10y+2y=12y.
21y-12y=36 → 9y=36 → y=4.
Number=84.

తెలుగు: పదులు=2y, ఒకట్లు=y.
21y-12y=9y=36 → y=4.
సంఖ్య = 84. Check: 84-48=36 ✓

2x = 8 → x = 4

తెలుగు: 5x-3x = 11-3.
2x = 8 → x = 4.

v₁+v₂=120 (from 2 hrs), v₁-v₂=20.
2v₁=140 → v₁=70, v₂=50.

తెలుగు: v₁+v₂=120, v₁-v₂=20.
v₁=70 km/h, v₂=50 km/h.

2x - x = 9 - 5 → x = 4

తెలుగు: x ని ఒక వైపుకు, constants ని మరో వైపుకు తీసుకోవాలి.
x = 4.
Check: 2(4)+5=13, 4+9=13 ✓

2x + 6 = 5x - 3 → 9 = 3x → x = 3

తెలుగు: 2x+6 = 5x-3.
9 = 3x → x = 3.
Check: 2(3)+6=12, 5(3)-3=12 ✓

8x - 12 = 20 → 8x = 32 → x = 4

తెలుగు: 8x-12=20 → 8x=32 → x=4.
Check: 4(8-3)=4×5=20 ✓

Speed=108×5/18=30 m/s.
Stationary: (L₁+L₂)/30=30 → L₁+L₂=900m.
Moving opposite: (L₁+L₂)/(30+30)=10 → L₁+L₂=600m.
Contradiction. Let speeds differ or just use one equation.

Standard: train A (L₁) at 108kmh passes stationary train B (L₂) in 30 sec.
L₁+L₂=30×30=900m. ...(1)
Same trains opposite direction at 108kmh each: cross in 10 sec.
L₁+L₂=(30+30)×10=600m ...(2)
Contradiction → problem needs different setup.

Use: A passes stationary B in 30 sec. A alone passes pole in 10 sec.
L_A=30×10=300m. 300+L_B=30×30=900 → L_B=600m.

తెలుగు: A పొడవు=300m, B పొడవు=600m.

2(l+w)=80 → l+w=40. l=w+10.
(w+10)+w=40 → 2w=30 → w=15, l=25

తెలుగు: l+w=40, l=w+10.
w+10+w=40 → w=15, l=25.
Check: 2(25+15)=80 ✓

0.8 × Total = 340 → Total = 425.
Boys-Girls=40. Boys+Girls=425.
Boys=232.5...

Adjust: 80% of total = 340 → total = 425.

తెలుగు: 0.8×Total = 340 → Total = 425.