Soustava dvou lineárních rovnic

Lineární rovnice [i]F(x,y)=0[/i] je obecnou rovnicí přímky v rovině. [br][list=1][*]Pokud všechny přímky procházejí jedním bodem, má soustava právě jedno řešení.[/*][*]Pokud mezi přímkami existuje dvojice rovnoběžných přímek, soustava nemá řešení. [/*][*]Pokud všechny rovnice vyjadřují tutéž přímku, má soustava nekonečně mnoho řešení.[br][/*][/list]
[b]Soustava 1:[/b][br]3x - y = 1 [br] x - y = -1[br]4x -2y = 0 [br]Matici soustavy převedeme ekvivalentními úpravami na horní trojúhelníkovou matici:[br][img]data:image/png;base64,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[/img][br]odtud y = 2 a x - y = -1, tedy řešením je uspořídaná dvojice (1,2).[br][br][b]Soustava 2:[/b][br]3x - y - 1 = 0[br]3x - y + 3 = 0[br][img]data:image/png;base64,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[/img][br]odtud 0y=-4, soustava nemá řešení[br][br][b]Soustava 3:[/b][br] x - y = -1[br] 3x - 3y = -3[br][br]Rovnice jsou lineárně závislé, řešením je jednoparametrický systém (t, t+1), [math]t\in R[/math].[br]
Řešení soustavy dvou lineárních rovnic o dvou neznámých
2x - y + 5 = 0[br]4x - 2y + 10 = 0
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Information: Soustava dvou lineárních rovnic