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Gi?i h? ph??ng tr¨¬nh m? 2 ?n 2 pt

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Gi?i h? ph??ng tr¨¬nh m? 2 ?n 2 pt Gi?i h? ph??ng tr¨¬nh m? sau ??y: ax + by = c a'x + b'y = c' Trong ?¨® a, b, c, a¡¯, b¡¯, c¡¯ l¨¤ c¨¢c s? m? cho tr??c; x v¨¤ y l¨¤ c¨¢c ?n s? m?. Gi?i h? ph??ng tr¨¬nh m? ax + by = c (1) a'x + b'y = c' (2) ---------- a = [la, ra] ; b = [lb, rb] ; c = [lc, rc] ; a¡¯ = [la¡¯, ra¡¯] ; b¡¯ = [lb¡¯, rb¡¯] ; c¡¯ = [lc¡¯, rc¡¯]; x = [lx, rx]; y = [ly, ry] ; Ta x¨¦t 64 tr??ng h?p : TH1- a>0, b>0, c>0, a¡¯>0, b¡¯>0, c¡¯>0 ¡­. TH64- a<0, b<0, c<0, a¡¯<0, b¡¯<0, c¡¯<0 ------------------------------- TH1- a>0, b>0, c>0, a¡¯>0, b¡¯>0, c¡¯>0 T¨¬m nghi?m x>0 v¨¤ y>0 (1) Ta c¨® : [ la, ra]*[lx, rx] + [ lb, rb]*[ly, ry] = [lc, rc] => [la.lx, ra.rx] + [lb.ly, rb.ry] = [lc, rc] => [la.lx + lb.ly, ra.rx + rb.ry] = [lc, rc] => la.lx + lb.ly = lc v¨¤ ra.rx + rb.ry = rc => lx = (lc ¨C lb.ly)/la v¨¤ rx = (rc ¨C rb.ry)/ra => x = [(lc ¨C lb.ly)/la, (rc ¨C rb.ry)/ra] (*) Thay (*) v¨¤o (2) ta c¨® : [ la¡¯, ra¡¯]*[ (lc ¨C lb.ly)/la, (rc ¨C rb.ry)/ra] + [ lb¡¯, rb¡¯]*[ly, ry] = [lc¡¯, rc¡¯] => [la¡¯. (lc ¨C lb.ly)/la], [ra¡¯. (rc ¨C rb.ry)/ra] + + [lb¡¯.ly, rb¡¯.ry] = [lc¡¯, rc¡¯] => [la¡¯. (lc ¨C lb.ly)/la + lb¡¯.ly, ra¡¯. (rc ¨C rb.ry)/ra + rb¡¯.ry] = [lc¡¯, rc¡¯] => la¡¯. (lc ¨C lb.ly)/la + lb¡¯.ly = lc¡¯ v¨¤ ra¡¯.(rc ¨C rb.ry)/ra + rb¡¯.ry = rc¡¯ => la¡¯.lc/la ¨C la¡¯.lb.ly/la + lb¡¯.ly = lc¡¯ v¨¤ ra¡¯.rc/ra ¨C ra¡¯.rb.ry/ra + rb¡¯.ry = rc¡¯ => ly(la.lb¡¯ ¨C la¡¯.lb) = la.lc¡¯ ¨C la¡¯.lc v¨¤ ry(ra.rb¡¯ ¨C ra¡¯.rb) = ra.rc¡¯ ¨C ra¡¯.rc => ly = (la.lc¡¯ ¨C la¡¯.lc)/ (la.lb¡¯ ¨C la¡¯.lb) v¨¤ ry = (ra.rc¡¯ ¨C ra¡¯.rc)/ (ra.rb¡¯ ¨C ra¡¯.rb) => y = [(la.lc¡¯ ¨C la¡¯.lc)/ (la.lb¡¯ ¨C la¡¯.lb), (ra.rc¡¯ ¨C ra¡¯.rc)/ (ra.rb¡¯ ¨C ra¡¯.rb)] V?y nghi?m h? ph??ng tr¨¬nh m? ax + by = c a'x + b'y = c' v?i x>0 v¨¤ y>0 l¨¤ :
x = [(lc ¨C lb.ly)/la, (rc ¨C rb.ry)/ra] y = [(la.lc¡¯ ¨C la¡¯.lc)/ (la.lb¡¯ ¨C la¡¯.lb), (ra.rc¡¯ ¨C ra¡¯.rc)/ (ra.rb¡¯ ¨C ra¡¯.rb)] Cu?i c¨´ng c?ng t¨¬m ???c b¨¤i gi?i, c?m ?n Anh tannv r?t nhi?u. Em ngh? m¨¬nh ch? x¨¦t a, b, a', b' th?i, v¨¬ c v¨¤ c' kh?ng tham v¨¤o ph¨¦p nh?n, n¨ºn kh?ng c?n x¨¦t. Do ?¨® ch? c¨°n: 16 tr??ng h?p Ta x¨¦t 16 tr??ng h?p : TH1- a>0, b>0, a¡¯>0, b¡¯>0 ¡­. TH16- a<0, b<0, a¡¯<0, b¡¯<0 Mong ACE ch? gi¨¢o th¨ºm L?u qu¨¢ m?i v¨¤o ??y, C¨¢m ?n anh t?n r?t nhi?u, N?u x¨¦t th¨ºm x , y n?a th¨¬ ph?i 64 X 4 tr??ng h?p. ??ng th?i c¨® 1 l?u ? l¨¤ lu?n lu?n s? d?ng ?i?u ki?n ban ??u v¨¤ left < right ?? th? v¨¤o c¨¢c gi¨¢ tr? ban ??u ?? c¨® ?i?u ki?n nghi?m. C¨® l? h?u n¨®i ch¨ªnh x¨¢c, v¨¤ theo b?n th?n m¨¬nh th¨¬ b?i v¨¬ b?n ch?t ph¨¦p nh?n v¨¤ ph¨¦p chia s? m? n¨® ph?c t?p n¨ºn ph?i chia tr??ng h?p => ch? n¨¤o c¨® ph¨¦p X,/ th¨¬ x¨¦t gi¨¢ tr? c?a c¨¢c bi?n s? v¨¤ tham s? >=<so v?i 0. ( chac kh?i x¨¦t = 0 v¨¬ alpha trong kho?ng (0,1] ). H¨¬nh nh? trong b¨¤i gi?ng c¨¢ch gi?i pt b?t 2 1 ?n s? c¨® s? d?ng delta , v?y t?i sao trong ??y ta kh?ng s? d?ng ??nh th?c ??nh Th?c sau khi x¨¦t a,b,a',b' ? Th? xem ====== ax + by = c (1) a'x + b'y = c' (2) ---------- a = [la, ra] ; b = [lb, rb] ; c = [lc, rc] ; a¡¯ = [la¡¯, ra¡¯] ; b¡¯ = [lb¡¯, rb¡¯] ; c¡¯ = [lc¡¯, rc¡¯]; x = [lx, rx]; y = [ly, ry] ; ------------------------------ D = ab' - a'b Dx = cb' - c'b Dy = ac' - a'c ---------------------------------------- TH 1/ a,b,a',b' > 0 xet nghiem x,y > 0
(1) <=> [la,ra]*[lx,rx] + [lb,rb]*[ly,ry] = [lc,rc] (2) <=> [la',ra']*[lx,rx] + [lb',rb']*[ly,ry] = [lc',rc'] <=> [la*lx+lb*ly,ra*rx+rb*ry] = [lc,rc] [la'*lx+lb'*ly,ra'*rx+rb'*ry] = [lc',rc'] th¨¤nh 2 h? lalx + lbly = lc la'lx + lb'ly = lc' (h1) v¨¤ rarx + rbry = rc ra'rx + rb'ry = rc' (h2) X¨¦t (h1) Dl = la*lb' - la'*lb Dxl = lc*lb' - lc'*lb Dyl = la*lc' - la'*lc xl = Dxl/Dl= (lc*lb' - lc'*lb)/(la*lb' - la'*lb) ?i?u ki?n : Dl <>0 : d?u <> l¨¤ d?u kh¨¢c v¨¤ xl > 0 => la*lb' <> la'*lb v¨¤ ( lc*lb'> lc'*lb v¨¤ la*lb' > la'*lb) ho?c ( lc*lb'< lc'*lb v¨¤ la*lb' < la'*lb) T??ng t? cho xr, yl, yr => Bi?n lu?n ?? r¨²t g?n c¨¢c ?i?u ki?n cho nh?, n?u ra ???c c, c' th¨¬ c¨¤ng t?t kh?ng th¨¬ ?? nguy¨ºn 1 b?y c?ng ch? sao. + nghi?m

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Gi?i h? ph??ng tr¨¬nh m? 2 ?n 2 pt

  • 1. Gi?i h? ph??ng tr¨¬nh m? 2 ?n 2 pt Gi?i h? ph??ng tr¨¬nh m? sau ??y: ax + by = c a'x + b'y = c' Trong ?¨® a, b, c, a¡¯, b¡¯, c¡¯ l¨¤ c¨¢c s? m? cho tr??c; x v¨¤ y l¨¤ c¨¢c ?n s? m?. Gi?i h? ph??ng tr¨¬nh m? ax + by = c (1) a'x + b'y = c' (2) ---------- a = [la, ra] ; b = [lb, rb] ; c = [lc, rc] ; a¡¯ = [la¡¯, ra¡¯] ; b¡¯ = [lb¡¯, rb¡¯] ; c¡¯ = [lc¡¯, rc¡¯]; x = [lx, rx]; y = [ly, ry] ; Ta x¨¦t 64 tr??ng h?p : TH1- a>0, b>0, c>0, a¡¯>0, b¡¯>0, c¡¯>0 ¡­. TH64- a<0, b<0, c<0, a¡¯<0, b¡¯<0, c¡¯<0 ------------------------------- TH1- a>0, b>0, c>0, a¡¯>0, b¡¯>0, c¡¯>0 T¨¬m nghi?m x>0 v¨¤ y>0 (1) Ta c¨® : [ la, ra]*[lx, rx] + [ lb, rb]*[ly, ry] = [lc, rc] => [la.lx, ra.rx] + [lb.ly, rb.ry] = [lc, rc] => [la.lx + lb.ly, ra.rx + rb.ry] = [lc, rc] => la.lx + lb.ly = lc v¨¤ ra.rx + rb.ry = rc => lx = (lc ¨C lb.ly)/la v¨¤ rx = (rc ¨C rb.ry)/ra => x = [(lc ¨C lb.ly)/la, (rc ¨C rb.ry)/ra] (*) Thay (*) v¨¤o (2) ta c¨® : [ la¡¯, ra¡¯]*[ (lc ¨C lb.ly)/la, (rc ¨C rb.ry)/ra] + [ lb¡¯, rb¡¯]*[ly, ry] = [lc¡¯, rc¡¯] => [la¡¯. (lc ¨C lb.ly)/la], [ra¡¯. (rc ¨C rb.ry)/ra] + + [lb¡¯.ly, rb¡¯.ry] = [lc¡¯, rc¡¯] => [la¡¯. (lc ¨C lb.ly)/la + lb¡¯.ly, ra¡¯. (rc ¨C rb.ry)/ra + rb¡¯.ry] = [lc¡¯, rc¡¯] => la¡¯. (lc ¨C lb.ly)/la + lb¡¯.ly = lc¡¯ v¨¤ ra¡¯.(rc ¨C rb.ry)/ra + rb¡¯.ry = rc¡¯ => la¡¯.lc/la ¨C la¡¯.lb.ly/la + lb¡¯.ly = lc¡¯ v¨¤ ra¡¯.rc/ra ¨C ra¡¯.rb.ry/ra + rb¡¯.ry = rc¡¯ => ly(la.lb¡¯ ¨C la¡¯.lb) = la.lc¡¯ ¨C la¡¯.lc v¨¤ ry(ra.rb¡¯ ¨C ra¡¯.rb) = ra.rc¡¯ ¨C ra¡¯.rc => ly = (la.lc¡¯ ¨C la¡¯.lc)/ (la.lb¡¯ ¨C la¡¯.lb) v¨¤ ry = (ra.rc¡¯ ¨C ra¡¯.rc)/ (ra.rb¡¯ ¨C ra¡¯.rb) => y = [(la.lc¡¯ ¨C la¡¯.lc)/ (la.lb¡¯ ¨C la¡¯.lb), (ra.rc¡¯ ¨C ra¡¯.rc)/ (ra.rb¡¯ ¨C ra¡¯.rb)] V?y nghi?m h? ph??ng tr¨¬nh m? ax + by = c a'x + b'y = c' v?i x>0 v¨¤ y>0 l¨¤ :
  • 2. x = [(lc ¨C lb.ly)/la, (rc ¨C rb.ry)/ra] y = [(la.lc¡¯ ¨C la¡¯.lc)/ (la.lb¡¯ ¨C la¡¯.lb), (ra.rc¡¯ ¨C ra¡¯.rc)/ (ra.rb¡¯ ¨C ra¡¯.rb)] Cu?i c¨´ng c?ng t¨¬m ???c b¨¤i gi?i, c?m ?n Anh tannv r?t nhi?u. Em ngh? m¨¬nh ch? x¨¦t a, b, a', b' th?i, v¨¬ c v¨¤ c' kh?ng tham v¨¤o ph¨¦p nh?n, n¨ºn kh?ng c?n x¨¦t. Do ?¨® ch? c¨°n: 16 tr??ng h?p Ta x¨¦t 16 tr??ng h?p : TH1- a>0, b>0, a¡¯>0, b¡¯>0 ¡­. TH16- a<0, b<0, a¡¯<0, b¡¯<0 Mong ACE ch? gi¨¢o th¨ºm L?u qu¨¢ m?i v¨¤o ??y, C¨¢m ?n anh t?n r?t nhi?u, N?u x¨¦t th¨ºm x , y n?a th¨¬ ph?i 64 X 4 tr??ng h?p. ??ng th?i c¨® 1 l?u ? l¨¤ lu?n lu?n s? d?ng ?i?u ki?n ban ??u v¨¤ left < right ?? th? v¨¤o c¨¢c gi¨¢ tr? ban ??u ?? c¨® ?i?u ki?n nghi?m. C¨® l? h?u n¨®i ch¨ªnh x¨¢c, v¨¤ theo b?n th?n m¨¬nh th¨¬ b?i v¨¬ b?n ch?t ph¨¦p nh?n v¨¤ ph¨¦p chia s? m? n¨® ph?c t?p n¨ºn ph?i chia tr??ng h?p => ch? n¨¤o c¨® ph¨¦p X,/ th¨¬ x¨¦t gi¨¢ tr? c?a c¨¢c bi?n s? v¨¤ tham s? >=<so v?i 0. ( chac kh?i x¨¦t = 0 v¨¬ alpha trong kho?ng (0,1] ). H¨¬nh nh? trong b¨¤i gi?ng c¨¢ch gi?i pt b?t 2 1 ?n s? c¨® s? d?ng delta , v?y t?i sao trong ??y ta kh?ng s? d?ng ??nh th?c ??nh Th?c sau khi x¨¦t a,b,a',b' ? Th? xem ====== ax + by = c (1) a'x + b'y = c' (2) ---------- a = [la, ra] ; b = [lb, rb] ; c = [lc, rc] ; a¡¯ = [la¡¯, ra¡¯] ; b¡¯ = [lb¡¯, rb¡¯] ; c¡¯ = [lc¡¯, rc¡¯]; x = [lx, rx]; y = [ly, ry] ; ------------------------------ D = ab' - a'b Dx = cb' - c'b Dy = ac' - a'c ---------------------------------------- TH 1/ a,b,a',b' > 0 xet nghiem x,y > 0
  • 3. (1) <=> [la,ra]*[lx,rx] + [lb,rb]*[ly,ry] = [lc,rc] (2) <=> [la',ra']*[lx,rx] + [lb',rb']*[ly,ry] = [lc',rc'] <=> [la*lx+lb*ly,ra*rx+rb*ry] = [lc,rc] [la'*lx+lb'*ly,ra'*rx+rb'*ry] = [lc',rc'] th¨¤nh 2 h? lalx + lbly = lc la'lx + lb'ly = lc' (h1) v¨¤ rarx + rbry = rc ra'rx + rb'ry = rc' (h2) X¨¦t (h1) Dl = la*lb' - la'*lb Dxl = lc*lb' - lc'*lb Dyl = la*lc' - la'*lc xl = Dxl/Dl= (lc*lb' - lc'*lb)/(la*lb' - la'*lb) ?i?u ki?n : Dl <>0 : d?u <> l¨¤ d?u kh¨¢c v¨¤ xl > 0 => la*lb' <> la'*lb v¨¤ ( lc*lb'> lc'*lb v¨¤ la*lb' > la'*lb) ho?c ( lc*lb'< lc'*lb v¨¤ la*lb' < la'*lb) T??ng t? cho xr, yl, yr => Bi?n lu?n ?? r¨²t g?n c¨¢c ?i?u ki?n cho nh?, n?u ra ???c c, c' th¨¬ c¨¤ng t?t kh?ng th¨¬ ?? nguy¨ºn 1 b?y c?ng ch? sao. + nghi?m