yugapat sameekaran

ganit ke sandarbh mein ek se adhik charon (variables or unknown) se yukt ek se adhik sameekaranon ke samooh ko yugapat sameekaran (simultaneous equations) kehte hain. 'yugapat' ka arth hai - 'ek saath'.

diye gaye sameekaranon par ganiteeya sankriyaayein karke unamein aane vaale sabhi charon ka aisa maan nikaalate hain jo sabhi sameekaranon ko santusht karein; in charon ka maan praapt karna hi yugapat sameekaran ka hal kahalaata hai.

udaaharan

nimnalikhit do sameekaran ek yugapat sameekaran ko niroopit karte hain. isame x aur y do char hain. is udaaharan mein diye hue dono sameekaran rekheeya beejeeya sameekaran hain. kintu yugapat sameekaranon ke samooh mein kuchh ya sabhi sameekaran arekhiiya ya abeejeeya (jaise trikonamittiya) ho sakte hain.

sameekaranon ke is nikaaya (system of equations) ka hal nimnalikhit hai:

isi prakaar, nimnalikhit sameekaran nikaaya ka hal hai - x = 100, y = 0

anukram

yugapat sameekaranon ka mahatva

vigyaan, takaneeki evam anya kshetron mein kisi samasya ka vishleshan karte samay ya kisi design ke kaaryaanvayan mein aksar yugapat sameekaranon ka nikaaya hal karna padata hai. iske alaava vibhinn ganiteeya samasyaaon ka hal nikaalate samay bhi ant mein yugapat sameekaran praapt ho sakta hai.

yugapat sameekaranon ka hal

yugapat sameekaranon ke hal ke liye praaya: nimnalikhit vidhiyaaain upayog mein laayi jaati hain:

  • pratisthaapan vidhi (substitution)
  • graafeeya vidhi (sameekaranon ka graaf kheenchakar)
  • vilopan vidhi (elimination)
  • maitriks vidhi
  • aankik vidhi (nyumerikl methad)

pratisthaapan vidhi

is vidhi mein diye gaye sameekaranon ke nikaaya mein se koi ek sameekaran lete hain aur usako is roop mein likhte hain ki koi agyaat raashi anya agyaat raashiyon ke falan ke roop mein abhivyakt ho jaaya. jaise, yadi mool sameekaran hai to ise ) ke roop mein likheinge. y ka yeh maan shesh sabhi sameekaranon mein rakh deinge. is prakaar hamaare paas pehle ki apeksha ek kam sameekaran bacha aur agyaat raashi bhi pehle ki apeksha ek kam ho gayi. yahi kram tab tak chalaate hain jab tak hamein keval ek sameekaran na mil jaaya. isse usamein nihit agyaat raashi ka maan nikaalate hain aur ulte kram mein (back substitution) pratisthaapan karte hue anya agyaat raashiyon ka maan gyaat kar liya jaata hai. neeche ke udaaharan se yeh spasht ho jaayega-

udaaharan:

inmein sabse pehle ko hataane ki koshish karte hain-

ab doosare sameekaran se x ka maan y ke falan ke roop mein vyakt karte hue:

thoda saral karne par y ka maan nikal jaata hai:

ab ulte kram mein pehle x aur fir z ka maan nikaalate hain-

'