三元二次方程组求解

三元二次方程组的解法~

a+b=5ab
a+c=6ac
b+c=7bc

a+b=5ab, a+b-5ab=0, a(1-5b)=-b, a=b/(5b-1)
b+c=7bc, 7bc-c=b, c(7b-1)=b, c=b/(7b-1)
把a=b/(5b-1)，c=b/(7b-1)代入：a+c=6ac， b/(5b-1)+b/(7b-1)=6[b/(5b-1)]*[b/(7b-1)]
1/(5b-1)+1/(7b-1)=6b/[(5b-1)(7b-1)]
12b-2=6b
6b=2
b=1/3
a=b/(5b-1)=(1/3)/(5/3-1)=(1/3)/(2/3)=1/2
c=b/(7b-1)=(1/3)/(7/3-1)=(1/3)/(4/3)=1/4

解题思路：
a、b、c三个未知数，把其中2个未知数用剩余的一个未知数来表示出来，
代入原试题中，就变成了一个未知数的方程去解。

x1=-226.11;
y1=238.802;
z1=-1577;
x2=223.1317;
y2=201.5533;
z2=-1577;
x3=-277.2719;
y3=-202.0605;
z3=-1577;
syms t1 t2 t3
eq1=(t1*x1-t2*x2)^2+(t1*y1-t2*y2)^2+(t1*z1-t2*z2)^2-532.8689^2;
eq2=(t1*x1-t3*x3)^2+(t1*y1-t3*y3)^2+(t1*z1-t3*z3)^2-532.8689^2;
eq3=(t3*x3-t2*x2)^2+(t3*y3-t2*y2)^2+(t3*z3-t2*z2)^2-753.5904^2;
[t1 t2 t3]=solve(eq1,eq2,eq3,'t1','t2','t3')
楼主我用solve可以解出解来，你参考一下

这样的方程无法求解出具体的解，只能用软件找近似解，我特意花了点时间用matlab给你算了下，解有好几种，我把程序和解拷在下面，你自己对应看吧，x,y,z相同行对应的是一组解：
>> [x,y,z] = solve('x^2 + y + z = 3','x + 0.5*y^2 + z = 2','z^2+1/3*x+z=1')

x =

-2.1558096397175413208632102682324
2.3417683016658045479995603587582
-1.2694054456821613794740015901516
1.5697981332259246563154829507059
1.6611588330789069259700525786493
-0.59358614561699753216141747743743
1.3994819294241634717847593430704
-2.9534059663780993695712258953623

y =

-2.5505844145482914103793970728305
-1.2987433423594474975158583431781
3.1821070694215204544101697788912
1.8882207449206230289046359054338
-0.093882841658254402986456722184425
1.9443840933491424819684799922912
0.65629157310122349342520959953843
-3.7277928822265161478267831379616

z =

0.90306921184921609706913053220585
-1.1851354363272990742614441731111
-1.7934972549490472188215316354913
-1.3524869240002209253211014457879
0.33443417294217863986017055022295
0.70327139438241411947041541668009
0.38515875611399723798553159318324
-1.9948139200112388759811708379018

这个很难给出精确解，我能找到的数值近似解只有这些
{x -> -3.04819, y -> -3.86614, z -> -2.42533}, {x -> -1.74924,
y -> -1.93567, z -> 1.87583}, {x -> -0.741927 - 0.314823i,
y -> 2.31845 + 0.593102 i,
z -> 0.23021 - 1.06025i}, {x -> -0.741927 + 0.314823i,
y -> 2.31845 - 0.593102i,
z -> 0.23021 + 1.06025i}, {x -> 1.21033 - 0.050519i,
y -> 1.10088 - 0.711429i,
z -> 0.436759 + 0.833718i}, {x -> 1.21033 + 0.050519i,
y -> 1.10088 + 0.711429i,
z -> 0.436759 - 0.833718i}, {x -> 1.50069, y -> 0.290933,
z -> 0.456985}, {x -> 2.35992, y -> -1.32779, z -> -1.24143}

[[x = -3*RootOf(81*_Z^8+324*_Z^7+162*_Z^6-630*_Z^5-423*_Z^4+522*_Z^3+175*_Z^2-208*_Z+38)^2-3*RootOf(81*_Z^8+324*_Z^7+162*_Z^6-630*_Z^5-423*_Z^4+522*_Z^3+175*_Z^2-208*_Z+38)+3, y = -9*RootOf(81*_Z^8+324*_Z^7+162*_Z^6-630*_Z^5-423*_Z^4+522*_Z^3+175*_Z^2-208*_Z+38)^4-18*RootOf(81*_Z^8+324*_Z^7+162*_Z^6-630*_Z^5-423*_Z^4+522*_Z^3+175*_Z^2-208*_Z+38)^3+9*RootOf(81*_Z^8+324*_Z^7+162*_Z^6-630*_Z^5-423*_Z^4+522*_Z^3+175*_Z^2-208*_Z+38)^2+17*RootOf(81*_Z^8+324*_Z^7+162*_Z^6-630*_Z^5-423*_Z^4+522*_Z^3+175*_Z^2-208*_Z+38)-6, z = RootOf(81*_Z^8+324*_Z^7+162*_Z^6-630*_Z^5-423*_Z^4+522*_Z^3+175*_Z^2-208*_Z+38)]]

这是我第一次见这么可怕的解。。
这是Algebraic Answer。。 RootOf就是当这个东西等于0的时候的值。。

我勒个去.........目测解完要死人了........

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三元二次方程组求解视频