What is Elimination method ?
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Friends, mera naam hai Dheeraj Sahni aur me apni iss website par aapko explain karta hoon maths se related complex topics in simple and easy way with solved examples !
Friends, maine aapko apne pervious article– What is Substitution method me bataya tha ki Substitution method kya hota hai aur uss article me hamne ye bhi seekha tha ki kisi Pair of Linear Equations in Two Variables ko Substitution method se kaise solve karte hai.
Friends, maine apne ek aur article– Pair of Linear Equations in Two Variables me Pair of Linear Equations in Two Variables ke baare me bataya hai aur uss article me ye bhi bataya hai ki kisi Pair of Linear Equations in Two Variables ko Graphical/Geometrical method se kaise solve karte hain.
Friends agar aap ye dono articles padhna chahte hain to neeche diye gaye link par click kijiye.
[What is Substitution method ? –iss article ko padhne ke liye blue link par click kijiye].
[Pair of Linear Equations in Two Variables –iss article ko padhne ke liye iss blue link par click kijiye].
Friends, me iss article mein aaj aapko batane waala hoon ki kisi Pair of Linear Equations in Two Variables ko Elimination method se kaise solve karte hain. Waise to friends kisi Pair of Linear Equations in Two Variables ko algebraic method se solve karne teen(three) tarike/methods hote hain jo ki maine aapko apne previous articles (ye dono articles jinke baare maine upar kaha hai) me pehle bhi bataya hai but iss article me hum unn teeno me se sirf ek hi method– Elimination method ke baare me discuss karenge. Iss article mein me aapko bataoonga ki Pair of Linear Equations in Two Variables ko Elimination method se kaise solve karte hain. So friends,
Step1: Sabse pehle aap uss Pair of Linear Equations in Two Variables me upasthit/present dono 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 suitable non-zero constants(real numbers) se multply karte hain aur wo constants(real numbers) aise hone chahiye ki multiply hone ke baad uss Pair of Linear Equations in Two Variables mein upasthit/present dono equations me kisi ek variable, ya toh x ke ya phir y ke coefficients barabar/equal ho jaaye(maan lijiye aapne y ke coefficients equal kar diye). Ab aapko ek naya/new Pair of Linear Equations in Two Variables milega. Ab aap uss naye Pair of Linear Equations in Two Variables me upasthit/present dono equations ko number dijiye. Dono me se kisi ek ko equation (3) aur doosre waale ko equation (4) maan lijiye.
Step3: Ab equation (3) aur equation (4) ko add ya subtract kar dijiye jisse ki equal coefficients wale terms(wo term jisme y hai) cancel/eliminate ho jaaye. Ab iss equation ko solve karke x ki value find kar lijiye. To hamne x ki value find kar li.
Step4: Ab x ki value kisi bhi equation, equation (1) ya phir (2) me rakh dijiye aur phir isse solve kijiye. Issse y ki value find ho jaayegi. To hamne y ki value bhi find kar li.
That's All !
Ab chaliye kuch examples ke jariye elimination method ko aur achchhe se samajhte hain...
Ex.1. Solve the following pair of linear equations in two variables by the elimination method–
9x–4y=2000
and 7x–3y=2000
Solution: Hame upar diye gaye elimination method ke steps follow karne hai–
Step1: 9x–4y=2000 -------(1)
7x–3y=2000 -------(2)
Step2: Now , multiplying equation (1) and equation (2) by 3 and 4 respectively to make the coefficients of y in both equations equal.
[Note:- Agar aapko ye pata nahi lag paa rha hai ki kis numbers se dono equations ko multiply karna hai to aapko ye karna hai– aap ye choose kijiye ki aapko kis variable ke( x ke ya phir y ke) coefficients equal karne hain( maan lijiye aapne y ko choose kar liya). Ye dicide karne ke baad aapko uss variable ke coefficients ka LCM lena hoga( aap 4 aur 3 ka LCM likaal lijiye jo ki 12 hai). Jitna LCM aata hai uske equal hi hame uss variable ko coefficients ko karna padta hai( matlab aapko dono equations me y ke coefficients ko 12 ke equal karna hai). Ab equation (1) me y ke coefficient se 12 ko divide kijiye; divide karke jo number aayega uss number se hi equation (1) ko multiply karna hoga( 12 ko 4 se divide karne par 3 aata hai to 3 se hi equation (1) ko multiply karna hoga). Isi tarah equation (2) me y ke coefficient se 12 ko divide kijiye; divide karke jo number aayega uss number se hi equation (2) ko multiply karna hoga( 12 ko 3 se divide karne par 4 aata hai to 4 se hi equation (2) ko multiply karna hoga)].
27x–12y=6000 -------(3)
28x–12y=8000 -------(4)
Step3: Subtracting equation (3) from equation (4)–
27x–12y = 6000
28x–12y = 8000
– + –
– x = – 2000
So, x= 2000
So, we have found the value of x.
Step4: Putting the value of x in equation (1),
9(2000)–4y=2000
18000–4y=2000
–4y=2000–18000
–4y=–16000
4y=16000
y=16000/4
y=4000
So, we have found the value of y also.
So, x=2000 and y=4000.
We are done !
Ex.2. Solve the following pair of linear equations in two variables by the elimination method–
3x–y=3
and 9x–3y=9
Solution: Following the steps–
Step1: 3x–y=3 ----------(1)
9x–3y=9 ----------(2)
Step2: Now , multiplying equation (1) and equation (2) by 3 and 1 respectively to make the coefficients of y in both equations equal.
9x–3y=9 ----------(3)
9x–3y=9 ----------(4)
Step3: Subtracting equation (3) from equation (4)–
9x – 3y = 9
9x – 3y = 9
– + –
0 = 0
We know, it is a true statement. So, this Pair of Linear Equations in Two Variables has infinitely many solutions.
[Note:- Agar kisi Pair of Linear Equations in Two Variables ko solve karte time agar hame koi true statement mile 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 elimination method–
2x+3y=8
and 4x+6y=7
Solution: Following the steps–
Step1: 2x+3y=8 ----------(1)
4x+6y=7 ----------(2)
Step2: Now , multiplying equation (1) and equation (2) by 2 and 1 respectively to make the coefficients of y in both equations equal.
4x+6y=8 ----------(3)
4x+6y=7 ----------(4)
Step3: Subtracting equation (3) from equation (4)–
4x + 6y = 8
4x + 6y = 7
– – –
0 = 1
We know, it is a false statement. So, this Pair of Linear Equations in Two Variables has no solution.
[Note:- Agar kisi Pair of Linear Equations in Two Variables ko solve karte time agar hame koi false statement mile to uss Pair of Linear Equations in Two Variables ka koi solution nahi hota hai].
We are done !
Ab me aapko ek aisa method/tarika bataoonga jiska use karke kisi Pair of Linear Equation in Two Variables ko bina solve kiye aap ye bata sakte ho ki uss Pair of Linear Equation in Two Variables ka ek unique solution hai ya infinitely many solutions hain ya phir koi solution nahi hai. Ye tarikaa/method me apne previous do(two), jinke baare mein maine upar kaha hai, articles me bhi bata chuka hoon !
Ab chaliye dekhte hain ki ye method kya hai aur ise kaise use karte 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-
I hope ki aapko sab kuch acche se samajh me aa gaya hoga. Aur agar aapko koi cheez samajh me nahi aayi ho to commemt ke through aap hame bata sakte hain. Agar aapko ye post accha laga ho to comment ke maadhyam se aap hame support kar sakte hai.
Aapne is article ko padhkar kya seekha?
CHECK YOUR KNOWLEDGE
Give answers:
Q.1-Is the Pair of Linear Equations in Two Variables 5x-4y+8=0 and 7x+6y–9=0 inconsistent?
Q.2-Solve the Pair of Linear Equations in Two Variables x+3y–12=0 and 8x–6y–4=0 by elimination method.
Q.3-Solve the Pair of Linear Equations in Two Variables 8x–3y+10=0 and 2x–5y+9=0 by elimination method.
Aap apne answers hame comment ke through bata sakte hain.
THANKS FOR READING THIS BLOG.
I hope everything is fine in your life and you are enjoying your day !
Friends, mera naam hai Dheeraj Sahni aur me apni iss website par aapko explain karta hoon maths se related complex topics in simple and easy way with solved examples !
Friends, maine aapko apne pervious article– What is Substitution method me bataya tha ki Substitution method kya hota hai aur uss article me hamne ye bhi seekha tha ki kisi Pair of Linear Equations in Two Variables ko Substitution method se kaise solve karte hai.
Friends, maine apne ek aur article– Pair of Linear Equations in Two Variables me Pair of Linear Equations in Two Variables ke baare me bataya hai aur uss article me ye bhi bataya hai ki kisi Pair of Linear Equations in Two Variables ko Graphical/Geometrical method se kaise solve karte hain.
Friends agar aap ye dono articles padhna chahte hain to neeche diye gaye link par click kijiye.
[What is Substitution method ? –iss article ko padhne ke liye blue link par click kijiye].
[Pair of Linear Equations in Two Variables –iss article ko padhne ke liye iss blue link par click kijiye].
Friends, me iss article mein aaj aapko batane waala hoon ki kisi Pair of Linear Equations in Two Variables ko Elimination method se kaise solve karte hain. Waise to friends kisi Pair of Linear Equations in Two Variables ko algebraic method se solve karne teen(three) tarike/methods hote hain jo ki maine aapko apne previous articles (ye dono articles jinke baare maine upar kaha hai) me pehle bhi bataya hai but iss article me hum unn teeno me se sirf ek hi method– Elimination method ke baare me discuss karenge. Iss article mein me aapko bataoonga ki Pair of Linear Equations in Two Variables ko Elimination method se kaise solve karte hain. So friends,
Let's begin...
Elimination method kisi pair of linear equations in two variables ko solve karne ka ek algebraic method hai. Iss method me hame equations me se kisi ek variable ko cancel/eliminate karna padta hai( isi wajah se iss method ko Elimination method kehte hain !).
To aaiyee ab jaante hain elimination method ke baare main...
Elimination method
Elimination method se kisi pair of linear equations in two variables ko solve karte samay ye steps follow karne hote hain:–
To aaiyee ab jaante hain elimination method ke baare main...
Elimination method
Elimination method se kisi pair of linear equations in two variables ko solve karte samay ye steps follow karne hote hain:–
Step1: Sabse pehle aap uss Pair of Linear Equations in Two Variables me upasthit/present dono 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 suitable non-zero constants(real numbers) se multply karte hain aur wo constants(real numbers) aise hone chahiye ki multiply hone ke baad uss Pair of Linear Equations in Two Variables mein upasthit/present dono equations me kisi ek variable, ya toh x ke ya phir y ke coefficients barabar/equal ho jaaye(maan lijiye aapne y ke coefficients equal kar diye). Ab aapko ek naya/new Pair of Linear Equations in Two Variables milega. Ab aap uss naye Pair of Linear Equations in Two Variables me upasthit/present dono equations ko number dijiye. Dono me se kisi ek ko equation (3) aur doosre waale ko equation (4) maan lijiye.
Step3: Ab equation (3) aur equation (4) ko add ya subtract kar dijiye jisse ki equal coefficients wale terms(wo term jisme y hai) cancel/eliminate ho jaaye. Ab iss equation ko solve karke x ki value find kar lijiye. To hamne x ki value find kar li.
Step4: Ab x ki value kisi bhi equation, equation (1) ya phir (2) me rakh dijiye aur phir isse solve kijiye. Issse y ki value find ho jaayegi. To hamne y ki value bhi find kar li.
That's All !
Ab chaliye kuch examples ke jariye elimination method ko aur achchhe se samajhte hain...
Ex.1. Solve the following pair of linear equations in two variables by the elimination method–
9x–4y=2000
and 7x–3y=2000
Solution: Hame upar diye gaye elimination method ke steps follow karne hai–
Step1: 9x–4y=2000 -------(1)
7x–3y=2000 -------(2)
Step2: Now , multiplying equation (1) and equation (2) by 3 and 4 respectively to make the coefficients of y in both equations equal.
[Note:- Agar aapko ye pata nahi lag paa rha hai ki kis numbers se dono equations ko multiply karna hai to aapko ye karna hai– aap ye choose kijiye ki aapko kis variable ke( x ke ya phir y ke) coefficients equal karne hain( maan lijiye aapne y ko choose kar liya). Ye dicide karne ke baad aapko uss variable ke coefficients ka LCM lena hoga( aap 4 aur 3 ka LCM likaal lijiye jo ki 12 hai). Jitna LCM aata hai uske equal hi hame uss variable ko coefficients ko karna padta hai( matlab aapko dono equations me y ke coefficients ko 12 ke equal karna hai). Ab equation (1) me y ke coefficient se 12 ko divide kijiye; divide karke jo number aayega uss number se hi equation (1) ko multiply karna hoga( 12 ko 4 se divide karne par 3 aata hai to 3 se hi equation (1) ko multiply karna hoga). Isi tarah equation (2) me y ke coefficient se 12 ko divide kijiye; divide karke jo number aayega uss number se hi equation (2) ko multiply karna hoga( 12 ko 3 se divide karne par 4 aata hai to 4 se hi equation (2) ko multiply karna hoga)].
27x–12y=6000 -------(3)
28x–12y=8000 -------(4)
Step3: Subtracting equation (3) from equation (4)–
27x–12y = 6000
28x–12y = 8000
– + –
– x = – 2000
So, x= 2000
So, we have found the value of x.
Step4: Putting the value of x in equation (1),
9(2000)–4y=2000
18000–4y=2000
–4y=2000–18000
–4y=–16000
4y=16000
y=16000/4
y=4000
So, we have found the value of y also.
So, x=2000 and y=4000.
We are done !
Ex.2. Solve the following pair of linear equations in two variables by the elimination method–
3x–y=3
and 9x–3y=9
Solution: Following the steps–
Step1: 3x–y=3 ----------(1)
9x–3y=9 ----------(2)
Step2: Now , multiplying equation (1) and equation (2) by 3 and 1 respectively to make the coefficients of y in both equations equal.
9x–3y=9 ----------(3)
9x–3y=9 ----------(4)
Step3: Subtracting equation (3) from equation (4)–
9x – 3y = 9
9x – 3y = 9
– + –
0 = 0
We know, it is a true statement. So, this Pair of Linear Equations in Two Variables has infinitely many solutions.
[Note:- Agar kisi Pair of Linear Equations in Two Variables ko solve karte time agar hame koi true statement mile 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 elimination method–
2x+3y=8
and 4x+6y=7
Solution: Following the steps–
Step1: 2x+3y=8 ----------(1)
4x+6y=7 ----------(2)
Step2: Now , multiplying equation (1) and equation (2) by 2 and 1 respectively to make the coefficients of y in both equations equal.
4x+6y=8 ----------(3)
4x+6y=7 ----------(4)
Step3: Subtracting equation (3) from equation (4)–
4x + 6y = 8
4x + 6y = 7
– – –
0 = 1
We know, it is a false statement. So, this Pair of Linear Equations in Two Variables has no solution.
[Note:- Agar kisi Pair of Linear Equations in Two Variables ko solve karte time agar hame koi false statement mile to uss Pair of Linear Equations in Two Variables ka koi solution nahi hota hai].
We are done !
Ab me aapko ek aisa method/tarika bataoonga jiska use karke kisi Pair of Linear Equation in Two Variables ko bina solve kiye aap ye bata sakte ho ki uss Pair of Linear Equation in Two Variables ka ek unique solution hai ya infinitely many solutions hain ya phir koi solution nahi hai. Ye tarikaa/method me apne previous do(two), jinke baare mein maine upar kaha hai, articles me bhi bata chuka hoon !
Ab chaliye dekhte hain ki ye method kya hai aur ise kaise use karte 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].
Iss article me maine aapko teen(three) algebraic methods me se sirf ek method ke baare me bataya hai aur ek algebraic method ke baare mein maine apne previous article me bataya hai. Baaki jo ek algebraic method bach gaya uske baare mein me apne next articles me bataaonga. To aapse phir mulaakaat hogi mere next article mein.
I hope ki aapko mera ye article pasand aaya hoga aur mujhe ummeed hai ki aapko Elimination method ke baare me kafi jaankari mili hogi.
Aapne is article ko padhkar kya seekha?
CHECK YOUR KNOWLEDGE
Give answers:
Q.1-Is the Pair of Linear Equations in Two Variables 5x-4y+8=0 and 7x+6y–9=0 inconsistent?
Q.2-Solve the Pair of Linear Equations in Two Variables x+3y–12=0 and 8x–6y–4=0 by elimination method.
Q.3-Solve the Pair of Linear Equations in Two Variables 8x–3y+10=0 and 2x–5y+9=0 by elimination method.
Aap apne answers hame comment ke through bata sakte hain.
THANKS FOR READING THIS BLOG.
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