What is Cross-multiplication method?

Sunday, February 11, 2018

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Kya aap seekhna chahte hain ki Cross-multiplication method se kisi Pair of Linear Equations in Two Variables ko kaise solve kiya jaata hai ? Agar aap ka jawaab haa/yes hai to aap iss waqt ek dum sahi jagah par hain !

Friends, aaj me aapko bataane waala hoon Cross-multiplication method ke baare mein. Friendse aaj aapko bataoonga ki Cross-multiplication method kise kehte hain aur saath hi saath me aapko kuch examples ke jariye ye bhi samjhaaoonga ki kisi Pair of Linear Equations in Two Variables ko Cross-multiplication method se kaise solve karte hain. To friends, jyada time waste naa karte hue chailye main topic par aate hain. So, friends...

Let's begin...

Friends aaj ka topic hai– Cross-multiplication method

Cross-multiplication method

Ye kisi Pair of Linear Equations in Two Variables ko solve karne ka ek algebraic method hai. Ye algebraic method bohot saare reasons ki wajah se kisi Pair of Linear Equations in Two Variables ko solve karne ka bohot hi useful method hai. Kisi Pair of Linear Equations in Two Variables ko Cross-multiplication method se solve karne ke liye ye steps follow karne padte hain–

Step1: Sabse pehle aap uss Pair of Linear Equations in Two Variables me upasthit/present dono equations ko Pair of Linear Equations in Two Variables ke general form me likhiye aur phir unn equations ko number dijiye. Dono me se kisi ek ko equation (1) aur doosre waale ko equation (2) maan lijiye.

Step2: Ab aap uss Pair of Linear Equations in Two Variables mein dono equation ko iss diagram ke form me likh lijiye:

Step3: Ab iss diagram ki help se diye gaye arrows ke according coefficients ko multiply karke iss form me likh lijiye:

( Shayad aapko ab samajh me aa gaya hoga ki iss method ko Cross-multiplication method kyu lehte hain. Iss method me coefficients ki cross multiply karni padti hai isiliye iss method ko Cross-multiplication method kehte hain).

Step4: Ab iss equation ke first part aur last part ko equal rakhkar x ki value find kar lijiye par dhyaan rahe ki a₁b₂–a₂b₁≠0 aur issi tarah iss equation ke middle part aur last part ko equal rakhkar y ki value find kar lijiye par dhyaan rahe ki a₁b₂–a₂b₁≠0.

Iss tarah dono variables (yaha par x aur y) ki value mil jaati hai.

That's All !

Chaliye ab kuch examples ke jariye Cross-multiplication method ko aur sahi tarike se samajhte hain...

Ex.1. Solve the following pair of linear equations in two variables by the Cross-multiplication method–
7x+2y=1
and 3y+1=8x

Solution: Hame upar diye gaye substitution method ke steps follow karne hai–

Step1:     7x+2y–1=0 -----------(1)
–8x+3y+1=0   -----------(2)

Step2: Now ,

x y 1
2 –1 7 2
3 1 –8 3

Step3:
x   = y   = 1
(2×1)–(3×–1) (–1×–8)–(1×7) (7×3)–(–8×2)

x    =   y   =   1
(2) – (–3) (8)–(7) (21)–(–16)

    x    =   y   =   1
2 + 3 8–7 21+ 16

    x    =   y   =   1
5 1 37

Step4:
   x    =   1
   5 37
37 × x = 1 × 5
37x = 5
x = 5/37

       y    =   1
   1 37
37 × y = 1 ×1
37y = 1
y = 1/37
So hamne x ki aur y ki, dono ki values find kar li.

We are done !

Ex.2. Solve the following pair of linear equations in two variables by the Cross-multiplication method–
2x+3=y
and 4x–2y+6=0

Solution: Hame upar diye gaye substitution method ke steps follow karne hai–

Step1:    2x–y+3=0 -----------(1)
  4x–2y+6=0   -----------(2)

Step2: Now ,

x y 1
–1 3 2 –1
–2 6 4 –2

Step3:
x   =   y   =   1
(–1×6)–(–2×3) (3×4)–(6×2) (2×–2)–(4×–1)

    x    =   y   =   1
(–6) – (–6) (12)–(12) (–4)–(–4)

    x    =   y   =   1
–6 + 6 12 – 12 –4 + 4

    x    =   y   =   1
0 0 0

Step4:
     x    =   1
   0 0
0 × x = 1 × 0
0x = 0
x = 0/0 (it is not defined).

     y    =   1
   0 0
0 × y = 1 × 0
0y = 0
y = 0/0 (it is not defined).

So hum x ki aur y ki values find nahi kar paaye. Aisa issliye ho rha hai kyu ki upar diye gaye equation me x , y aur 1 teeno ka denominator 0 hai. So, iss Pair of Linear Equations in Two Variables ke infinitely many solutions hain.

[Note:- Agar kisi Pair of Linear Equations in Two Variables ko Cross-multiplication method se solve karte time agar hame x , y aur 1 teeno ke denominator 0 mile yaani ki agar b₁c₂–b₂c₁=0 , c₁a₂–c₂a₁=0 aur a₁b₂–a₂b₁=0 ho to uss Pair of Linear Equations in Two Variables ke infinitely many solutions hote hain].

We are done !

Ex.3. Solve the following pair of linear equations in two variables by the cross-multiplication method–
x+3y+7=0
and 2x+6y=8

Solution: Following the steps–

Step1:      x+3y+7=0 -----------(1)
2x+6y–8=0   -----------(2)

Step2: Now ,

x y 1
3 7 1 3
6 –8 2 6

Step3:
    x   =   y   =   1
(3×–8)–(6×7) (7×2)–(–8×1) (1×6)–(2×3)

   x    =   y   =   1
(–24) – (42) (14)–(–8) (6)–(6)

    x    =   y   =   1
–24 – 42 14 + 8 6 – 6

    x    =   y   =   1
–66 22 0

Step4:
    x    =   1
   –66 0
0 × x = 1 × –66
0x = –66
x = –66/0 (it is not defined).

      y    =   1
   22 0
0 × y = 1 × 22
0y = 22
y = 22/0 (it is not defined).

So hum x ki aur y ki values find nahi kar paaye. Aisa issliye ho rha hai kyu ki upar diye gaye equation me 1 ka denominator 0 hai but x aur y ka denominator 0 nahi hai. So, iss Pair of Linear Equations in Two Variables ka koi solution nahi hai.

[Note:- Agar kisi Pair of Linear Equations in Two Variables ko Cross-multiplication method se solve karte time agar hame 1 ka denominator 0 mile but x aur y ka denominator 0 naa ho yaani ki agar a₁b₂–a₂b₁=0 but b₁c₂–b₂c₁≠ 0 and c₁a₂–c₂a₁≠ 0 ho to uss Pair of Linear Equations in Two Variables ka koi solution nahi hota hai].

We are done !

Kya aap kisi Pair of Linear Equations in Two Variables ko bina solve kiye, sirf uss Pair of Linear Equations in Two Variables ko dekhkar ye bata sakte hain ki uss Pair of Linear Equations in Two Variables ka ek(unique) solution hai ya infinitely many solutions hain ya phir koi solution nahi hai ?

Chaliye me aapko batata hoon aap ke kaise pata kar sakte hain...

Sabse pehle hum diye gaye Pair of Linear Equation in Two Variables me a₁/a₂ aur b₁/b₂ nikaal lete hain. Agar ye dono equal nahi hai to uss Pair of Linear Equation in Two Variables ka graph intersecting lines hoga aur iss tarah iss situation me sirf ek(one) unique solution hoga. Agar a₁/a₂ aur  b₁/b₂  equal hain to hum c₁/c₂ nikaalte hain. Agar c₁/c₂  , a₁/a₂  aur b₁/b₂  ke equal hai to uss Pair of Linear Equation in Two Variables ka graph coincident lines hoga aur iss tarah iss situation me infinitely many solutions honge. Aur agar c₁/c₂  , a₁/a₂ aur  b₁/b₂  ke equal nahi hai to uss Pair of Linear Equation in Two Variables ka graph parallel lines hoga aur iss tarah iss situation me koi solution nahi hoga.
Iss table ko dekhkar aap iss method ko acche se samajh jaayenge-

[Note:- Jin Pair of Linear Equations in Two Variables ka solution hota hai chahe wo ek ho ya infinitely many, unhe consistent kehte hai aur jin Pair of Linear Equations in Two Variables ka koi bhi solution nahi hota usse inconsistent kehte hain].