Sat Jan 31 10:55:11 MST 1998 euler% math Mathematica 3.0 for Solaris Copyright 1988-96 Wolfram Research, Inc. -- Terminal graphics initialized -- In[1]:= In[2]:= In[3]:= (* ----------[ M a t h e m a t i c a ]---------- *) 0. Second In[4]:= (* ---------- Initialization ---------- *) 0. Second In[5]:= (* ---------- Equations ---------- *) 0. Second In[6]:= (* Manipulate an equation using a natural syntax: (x = 2)/2 + (1 = 1) => x/2 + 1 = 2 *) 0. Second In[7]:= (x == 2)/2 + (1 == 1) 0. Second x == 2 Out[7]= True + ------ 2 In[8]:= Map[#/2 + 1 &, x == 2] 0. Second x Out[8]= 1 + - == 2 2 In[9]:= (* Solve various nonlinear equations---this cubic polynomial has all\ > real roots *) 0. Second In[10]:= Solve[3*x^3 - 18*x^2 + 33*x - 19 == 0, x] 0.1 Second 3 + I Sqrt[3] 1/3 (-------------) 2 3 (3 + I Sqrt[3]) -(1/3) Out[10]= {{x -> 2 + ------------------ + (-----------------) }, 2/3 2 3 3 + I Sqrt[3] 1/3 (1 + I Sqrt[3]) (-------------) 2 > {x -> 2 - ---------------------------------- - 2/3 2 3 1 - I Sqrt[3] > ---------------------------}, 2/3 1/3 2 (3 (3 + I Sqrt[3])) 3 + I Sqrt[3] 1/3 (1 - I Sqrt[3]) (-------------) 2 > {x -> 2 - ---------------------------------- - 2/3 2 3 1 + I Sqrt[3] > ---------------------------}} 2/3 1/3 2 (3 (3 + I Sqrt[3])) In[11]:= Map[#[[1]] -> ComplexExpand[#[[2]]] &, Flatten[%]] 0.4 Second Pi Pi 2 Cos[--] Cos[--] 18 18 Pi Out[11]= {x -> 2 + ---------, x -> 2 - ------- + Sin[--], Sqrt[3] Sqrt[3] 18 Pi Cos[--] 18 Pi > x -> 2 - ------- - Sin[--]} Sqrt[3] 18 In[12]:= (* Some simple seeming problems can have messy answers: x = { [sqrt(5) - 1]/4 +/- 5^(1/4) sqrt(sqrt(5) + 1)/[2 sqrt(2)] i, - [sqrt(5) + 1]/4 +/- 5^(1/4) sqrt(sqrt(5) - 1)/[2 sqrt(2)] i} *) 0. Second In[13]:= eqn = x^4 + x^3 + x^2 + x + 1 == 0 0. Second 2 3 4 Out[13]= 1 + x + x + x + x == 0 In[14]:= Solve[eqn, x] 0. Second 1/5 2/5 3/5 4/5 Out[14]= {{x -> -(-1) }, {x -> (-1) }, {x -> -(-1) }, {x -> (-1) }} In[15]:= Map[#[[1]] -> ComplexExpand[#[[2]]] &, Flatten[%]] 0.08 Second 1 Sqrt[5] I 5 - Sqrt[5] Out[15]= {x -> -(-) - ------- - - Sqrt[-----------], 4 4 2 2 1 Sqrt[5] I 5 + Sqrt[5] > x -> -(-) + ------- + - Sqrt[-----------], 4 4 2 2 1 Sqrt[5] I 5 + Sqrt[5] > x -> -(-) + ------- - - Sqrt[-----------], 4 4 2 2 1 Sqrt[5] I 5 - Sqrt[5] > x -> -(-) - ------- + - Sqrt[-----------]} 4 4 2 2 In[16]:= ( (* Check one of the answers *) eqn /. %[[1]] ) $MaxExtraPrecision::meprec: $MaxExtraPrecision = 50. reached while evaluating 3 1 I 5 - Sqrt[5] 4 - + <<4>> + (-(-) + <<1>> - - Sqrt[-----------]) . Increasing the value of 4 4 2 2 $MaxExtraPrecision may help resolve the uncertainty. 0.05 Second 3 Sqrt[5] I 5 - Sqrt[5] Out[16]= - - ------- - - Sqrt[-----------] + 4 4 2 2 1 Sqrt[5] I 5 - Sqrt[5] 2 > (-(-) - ------- - - Sqrt[-----------]) + 4 4 2 2 1 Sqrt[5] I 5 - Sqrt[5] 3 > (-(-) - ------- - - Sqrt[-----------]) + 4 4 2 2 1 Sqrt[5] I 5 - Sqrt[5] 4 > (-(-) - ------- - - Sqrt[-----------]) == 0 4 4 2 2 In[17]:= Simplify[%] 0.05 Second Out[17]= True In[18]:= Clear[eqn] 0. Second In[19]:= (* x = {2^(1/3) +- sqrt(3), +- sqrt(3) - 1/2^(2/3) +- i\ > sqrt(3)/2^(2/3)} [Mohamed Omar Rayes] *) 0. Second In[20]:= Solve[x^6 - 9*x^4 - 4*x^3 + 27*x^2 - 36*x - 23 == 0, x] 0.06 Second 2 3 4 6 Out[20]= {{x -> Root[-23 - 36 #1 + 27 #1 - 4 #1 - 9 #1 + #1 & , 1]}, 2 3 4 6 > {x -> Root[-23 - 36 #1 + 27 #1 - 4 #1 - 9 #1 + #1 & , 2]}, 2 3 4 6 > {x -> Root[-23 - 36 #1 + 27 #1 - 4 #1 - 9 #1 + #1 & , 3]}, 2 3 4 6 > {x -> Root[-23 - 36 #1 + 27 #1 - 4 #1 - 9 #1 + #1 & , 4]}, 2 3 4 6 > {x -> Root[-23 - 36 #1 + 27 #1 - 4 #1 - 9 #1 + #1 & , 5]}, 2 3 4 6 > {x -> Root[-23 - 36 #1 + 27 #1 - 4 #1 - 9 #1 + #1 & , 6]}} In[21]:= ToRadicals[%] 0.23 Second 2 3 4 6 Out[21]= {{x -> Root[-23 - 36 #1 + 27 #1 - 4 #1 - 9 #1 + #1 & , 1]}, 2 3 4 6 > {x -> Root[-23 - 36 #1 + 27 #1 - 4 #1 - 9 #1 + #1 & , 2]}, 2 3 4 6 > {x -> Root[-23 - 36 #1 + 27 #1 - 4 #1 - 9 #1 + #1 & , 3]}, 2 3 4 6 > {x -> Root[-23 - 36 #1 + 27 #1 - 4 #1 - 9 #1 + #1 & , 4]}, 2 3 4 6 > {x -> Root[-23 - 36 #1 + 27 #1 - 4 #1 - 9 #1 + #1 & , 5]}, 2 3 4 6 > {x -> Root[-23 - 36 #1 + 27 #1 - 4 #1 - 9 #1 + #1 & , 6]}} In[22]:= (* x = {1, e^(+- 2 pi i/7), e^(+- 4 pi i/7), e^(+- 6 pi i/7)} *) 0. Second In[23]:= Solve[x^7 - 1 == 0, x] 0. Second 1/7 2/7 3/7 Out[23]= {{x -> 1}, {x -> -(-1) }, {x -> (-1) }, {x -> -(-1) }, 4/7 5/7 6/7 > {x -> (-1) }, {x -> -(-1) }, {x -> (-1) }} In[24]:= (* x = 1 +- sqrt(+-sqrt(+-4 sqrt(3) - 3) - 3)/sqrt(2) [Richard\ > Liska] *) 0. Second In[25]:= Solve[x^8 - 8*x^7 + 34*x^6 - 92*x^5 + 175*x^4 - 236*x^3 + 226*x^2 -\ > 140*x + 46 == 0, x] 0.4 Second 2 - I Sqrt[2 (3 - Sqrt[-3 + 4 Sqrt[3]])] Out[25]= {{x -> ----------------------------------------}, 2 2 + I Sqrt[2 (3 - Sqrt[-3 + 4 Sqrt[3]])] > {x -> ----------------------------------------}, 2 2 - I Sqrt[2 (3 + Sqrt[-3 + 4 Sqrt[3]])] > {x -> ----------------------------------------}, 2 2 + I Sqrt[2 (3 + Sqrt[-3 + 4 Sqrt[3]])] > {x -> ----------------------------------------}, 2 2 - Sqrt[2 (-3 - I Sqrt[3 + 4 Sqrt[3]])] > {x -> ----------------------------------------}, 2 2 + Sqrt[2 (-3 - I Sqrt[3 + 4 Sqrt[3]])] > {x -> ----------------------------------------}, 2 2 - Sqrt[2 (-3 + I Sqrt[3 + 4 Sqrt[3]])] > {x -> ----------------------------------------}, 2 2 + Sqrt[2 (-3 + I Sqrt[3 + 4 Sqrt[3]])] > {x -> ----------------------------------------}} 2 In[26]:= (* The following equations have an infinite number of solutions (let\ > n be an arbitrary integer): x = {log(sqrt(z) - 1), log(sqrt(z) + 1) + i pi} [+ n 2 pi i, + n 2 pi i]\ > *) 0. Second In[27]:= E^(2*x) + 2*E^x + 1 == z 0. Second x 2 x Out[27]= 1 + 2 E + E == z In[28]:= Simplify[Solve[%, x]] Solve::ifun: Inverse functions are being used by Solve, so some solutions may not be found. 0.26 Second Out[28]= {{x -> Log[-1 - Sqrt[z]]}, {x -> Log[-1 + Sqrt[z]]}} In[29]:= (* x = (1 +- sqrt(9 - 8 n pi i))/2. Real solutions correspond to n\ > = 0 => x = {-1, 2} *) 0. Second In[30]:= Solve[Exp[2 - x^2] == Exp[-x], x] Solve::ifun: Inverse functions are being used by Solve, so some solutions may not be found. 0.29 Second Out[30]= {{x -> -1}, {x -> 2}} In[31]:= (* x = -W[n](-1) [e.g., -W[0](-1) = 0.31813 - 1.33724 i] where\ > W[n](x) is the nth branch of Lambert's W function *) 0. Second In[32]:= Solve[Exp[x] == x, x] InverseFunction::ifun: Warning: Inverse functions are being used. Values may be lost for multivalued inverses. Solve::ifun: Inverse functions are being used by Solve, so some solutions may not be found. General::stop: Further output of Solve::ifun will be suppressed during this calculation. 0.23 Second Out[32]= {{x -> -ProductLog[-1]}} In[33]:= (* x = {-1, 1} *) 0. Second In[34]:= Solve[x^x == x, x] Solve::verif: Potential solution {x -> 0} cannot be verified automatically. Verification may require use of limits. 0.05 Second Out[34]= {{x -> 1}} In[35]:= (* This equation is already factored and so *should* be easy to\ > solve: x = {-1, 2*{+-arcsinh(1) i + n pi}, 3*{pi/6 + n pi/3}} *) 0. Second In[36]:= (x + 1) * (Sin[x]^2 + 1)^2 * Cos[3*x]^3 == 0 0. Second 3 2 2 Out[36]= (1 + x) Cos[3 x] (1 + Sin[x] ) == 0 In[37]:= Solve[%, x] 0.12 Second -Pi Pi Out[37]= {{x -> -1}, {x -> ---}, {x -> --}, {x -> -I ArcSinh[1]}, 6 6 > {x -> I ArcSinh[1]}} In[38]:= (* x = pi/4 [+ n pi] *) 0. Second In[39]:= Solve[Sin[x] == Cos[x], x] 0.05 Second -3 Pi Pi Out[39]= {{x -> -----}, {x -> --}} 4 4 In[40]:= Solve[Tan[x] == 1, x] 0.05 Second Pi Out[40]= {{x -> --}} 4 In[41]:= (* x = {pi/6, 5 pi/6} [ + n 2 pi, + n 2 pi ] *) 0. Second In[42]:= Solve[Sin[x] == 1/2, x] 0.03 Second Pi Out[42]= {{x -> --}} 6 In[43]:= (* x = {0, 0} [+ n pi, + n 2 pi] *) 0. Second In[44]:= Solve[Sin[x] == Tan[x], x] 0.07 Second Out[44]= {{x -> 0}} In[45]:= (* x = {0, 0, 0} *) 0. Second In[46]:= Solve[ArcSin[x] == ArcTan[x], x] 0.27 Second Out[46]= {{x -> 0}} In[47]:= (* x = sqrt[(sqrt(5) - 1)/2] *) 0. Second In[48]:= Solve[ArcCos[x] == ArcTan[x], x] 0.23 Second Out[48]= {{x -> Log[5] Log[5] -1 + Cosh[------] + Sinh[------] 2 2 Log[--------------------------------] 2 > Cosh[-------------------------------------] + 2 Log[5] Log[5] -1 + Cosh[------] + Sinh[------] 2 2 Log[--------------------------------] 2 > Sinh[-------------------------------------]}} 2 In[49]:= FullSimplify[%] 2.36 Second -1 + Sqrt[5] -1 + Sqrt[5] Log[------------] Log[------------] 2 2 Out[49]= {{x -> Cosh[-----------------] + Sinh[-----------------]}} 2 2 In[50]:= (* x = 2 *) 0. Second In[51]:= Solve[(x - 2)/x^(1/3) == 0, x] 0.03 Second Out[51]= {{x -> 2}} In[52]:= (* This equation has no solutions *) 0. Second In[53]:= Solve[Sqrt[x^2 + 1] == x - 2, x] 0.03 Second Out[53]= {} In[54]:= (* x = 1 *) 0. Second In[55]:= Solve[x + Sqrt[x] == 2, x] 0.03 Second Out[55]= {{x -> 1}} In[56]:= (* x = 1/16 *) 0. Second In[57]:= Solve[2*Sqrt[x] + 3*x^(1/4) - 2 == 0, x] 0.03 Second 1 Out[57]= {{x -> --}} 16 In[58]:= (* x = {sqrt[(sqrt(5) - 1)/2], -i sqrt[(sqrt(5) + 1)/2]} *) 0. Second In[59]:= Solve[x == 1/Sqrt[1 + x^2], x] 0.11 Second -1 + Sqrt[5] 1 + Sqrt[5] Out[59]= {{x -> Sqrt[------------]}, {x -> -I Sqrt[-----------]}} 2 2 In[60]:= (* This problem is from a computational biology talk => 1 - log_2[m\ > (m - 1)] *) 0. Second In[61]:= Solve[Binomial[m, 2]*2^k == 1, k] 0.28 Second -2 Log[---------] (1 - m) m Out[61]= {{k -> --------------}} Log[2] In[62]:= (* x = log(c/a) / log(b/d) for a, b, c, d != 0 and b, d != 1 [Bill\ > Pletsch] *) 0. Second In[63]:= Solve[a*b^x == c*d^x, x] 0.29 Second a Log[-] c Out[63]= {{x -> ------}} Log[d] In[64]:= (* x = {1, e^4} *) 0. Second In[65]:= Solve[Sqrt[Log[x]] == Log[Sqrt[x]], x] 0.06 Second 4 Out[65]= {{x -> 1}, {x -> E }} In[66]:= (* Recursive use of inverses, including multiple branches of rational fractional powers [Richard Liska] => x = +-(b + sin(1 + cos(1/e^2)))^(3/2) *) 0. Second In[67]:= Solve[Log[ArcCos[ArcSin[x^(2/3) - b] - 1]] + 2 == 0, x] 0.15 Second Out[67]= {{x -> 3 2 -2 -2 2 > -Sqrt[b + 3 b Sin[1 + Cos[E ]] + 3 b Sin[1 + Cos[E ]] + -2 3 > Sin[1 + Cos[E ]] ]}, 3 2 -2 -2 2 > {x -> Sqrt[b + 3 b Sin[1 + Cos[E ]] + 3 b Sin[1 + Cos[E ]] + -2 3 > Sin[1 + Cos[E ]] ]}} In[68]:= Simplify[%] 0.2 Second -2 3 Out[68]= {{x -> -Sqrt[(b + Sin[1 + Cos[E ]]) ]}, -2 3 > {x -> Sqrt[(b + Sin[1 + Cos[E ]]) ]}} In[69]:= (* x = {-0.784966, -0.016291, 0.802557} From Metha Kamminga-van\ > Hulsen, ``Hoisting the Sails and Casting Off with Maple'', _Computer Algebra Nederland Nieuwsbrief_, Number 13, December 1994, ISSN 1380-1260, 27--40.\ > *) 0. Second In[70]:= eqn = 5*x + Exp[(x - 5)/2] == 8*x^3 0. Second (-5 + x)/2 3 Out[70]= E + 5 x == 8 x In[71]:= Solve[eqn, x] Solve::tdep: The equations appear to involve transcendental functions of the variables in an essentially non-algebraic way. 0.83 Second (-5 + x)/2 3 Out[71]= Solve[E + 5 x == 8 x , x] In[72]:= FindRoot[eqn, {x, -0.75}] 0.01 Second Out[72]= {x -> -0.784966} In[73]:= FindRoot[eqn, {x, 0}] 0. Second Out[73]= {x -> -0.0162907} In[74]:= FindRoot[eqn, {x, 0.75}] 0.01 Second Out[74]= {x -> 0.802557} In[75]:= Clear[eqn] 0. Second In[76]:= (* x = {-1, 3} *) 0. Second In[77]:= Solve[Abs[x - 1] == 2, x] 0.01 Second Out[77]= {{x -> -1}, {x -> 3}} In[78]:= (* x = {-1, -7} *) 0. Second In[79]:= Solve[Abs[2*x + 5] == Abs[x - 2], x] 0.02 Second Out[79]= {{x -> -7}, {x -> -1}} In[80]:= (* x = +-3/2 *) 0. Second In[81]:= Solve[1 - Abs[x] == Max[-x - 2, x - 2], x] 0.02 Second 3 3 Out[81]= {{x -> -(-)}, {x -> -}} 2 2 In[82]:= (* x = {-1, 3} *) 0. Second In[83]:= Solve[Max[2 - x^2, x] == Max[-x, x^3/9], x] 0.14 Second Out[83]= {{x -> -1}, {x -> 3}} In[84]:= (* x = {+-3, -3 [1 + sqrt(3) sin t + cos t]} = {+-3, -1.554894} where t = (arctan[sqrt(5)/2] - pi)/3. The third answer is the root of x^3 + 9 x^2 - 18 = 0 in the interval (-2, -1). *) 0. Second In[85]:= solns = Solve[Max[2 - x^2, x] == x^3/9, x] 0.1 Second Out[85]= {{x -> -3}, {x -> 3}, {x -> 2/3 1/3 3 (1 - I Sqrt[3]) 3 (1 + I Sqrt[3]) (-2 + I Sqrt[5]) > -3 - --------------------- - ----------------------------------------}} -2 + I Sqrt[5] 1/3 2 2 (--------------) 3 In[86]:= FullSimplify[%] 6.67 Second 2 3 Out[86]= {{x -> -3}, {x -> 3}, {x -> Root[-18 + 9 #1 + #1 & , 2]}} In[87]:= N[%] 0. Second Out[87]= {{x -> -3.}, {x -> 3.}, {x -> -1.55489}} In[88]:= Map[#[[1]] -> ComplexExpand[#[[2]]] &, Flatten[solns]] 0.12 Second Out[88]= {x -> -3, x -> 3, x -> Sqrt[5] Sqrt[5] Pi - ArcTan[-------] Pi - ArcTan[-------] 2 2 > -3 - 3 Cos[--------------------] + 3 Sqrt[3] Sin[--------------------]} 3 3 In[89]:= Clear[solns] 0. Second In[90]:= (* z = 2 + 3 i *) 0. Second In[91]:= eqn = (1 + I)*z + (2 - I)*Conjugate[z] == -3*I 0. Second Out[91]= (1 + I) z + (2 - I) Conjugate[z] == -3 I In[92]:= Solve[eqn, z] InverseFunction::ifun: Warning: Inverse functions are being used. Values may be lost for multivalued inverses. Solve::tdep: The equations appear to involve transcendental functions of the variables in an essentially non-algebraic way. 0.02 Second Out[92]= Solve[(1 + I) z + (2 - I) Conjugate[z] == -3 I, z] In[93]:= eqn /. z -> x + I*y 0. Second Out[93]= (1 + I) (x + I y) + (2 - I) Conjugate[x + I y] == -3 I In[94]:= Solve[%, {x, y}] InverseFunction::ifun: Warning: Inverse functions are being used. Values may be lost for multivalued inverses. General::stop: Further output of InverseFunction::ifun will be suppressed during this calculation. Solve::tdep: The equations appear to involve transcendental functions of the variables in an essentially non-algebraic way. General::stop: Further output of Solve::tdep will be suppressed during this calculation. 0.03 Second Out[94]= Solve[(1 + I) (x + I y) + (2 - I) Conjugate[x + I y] == -3 I, {x, y}] In[95]:= ComplexExpand[eqn /. z -> x + I*y] 0.02 Second Out[95]= 3 x - (2 + I) y == -3 I In[96]:= Solve[%, {x, y}] Solve::svars: Equations may not give solutions for all "solve" variables. 0.01 Second 2 I Out[96]= {{x -> -I + (- + -) y}} 3 3 In[97]:= Clear[eqn] 0. Second In[98]:= (* => {f^(-1)(1), f^(-1)(-2)} assuming f is invertible *) 0. Second In[99]:= Solve[f[x]^2 + f[x] - 2 == 0, x] 0.04 Second (-1) (-1) Out[99]= {{x -> (f )[-2]}, {x -> (f )[1]}} In[100]:= Clear[eqns, vars] 0. Second In[101]:= (* Solve a 3 x 3 system of linear equations *) 0. Second In[102]:= eqn1 = x + y + z == 6 0. Second Out[102]= x + y + z == 6 In[103]:= eqn2 = 2*x + y + 2*z == 10 0. Second Out[103]= 2 x + y + 2 z == 10 In[104]:= eqn3 = x + 3*y + z == 10 0. Second Out[104]= x + 3 y + z == 10 In[105]:= (* Note that the solution is parametric: x = 4 - z, y = 2 *) 0. Second In[106]:= Solve[{eqn1, eqn2, eqn3}, {x, y, z}] Solve::svars: Equations may not give solutions for all "solve" variables. 0. Second Out[106]= {{x -> 4 - z, y -> 2}} In[107]:= (* A linear system arising from the computation of a truncated\ > power series solution to a differential equation. There are 189 equations to be solved for 49 unknowns. 42 of the equations are repeats of other equations; many others are trivial. Solving this system directly by Gaussian elimination is *not* a good idea. Solving the easy equations first is probably a\ > better method. The solution is actually rather simple. [Stanly Steinberg] => k1 = ... = k22 = k24 = k25 = k27 = ... = k30 = k32 = k33 = k35 = ... = k38 = k40 = k41 = k44 = ... = k49 = 0, k23 = k31 = k39, k34 = b/a k26, k42 = c/a k26, {k23, k26, k43} are arbitrary *) 0. Second In[108]:= eqns = { -b*k8/a+c*k8/a == 0, -b*k11/a+c*k11/a == 0, -b*k10/a+c*k10/a+k2 == 0, -k3-b*k9/a+c*k9/a == 0, -b*k14/a+c*k14/a == 0, -b*k15/a+c*k15/a == 0, -b*k18/a+c*k18/a-k2 == 0, -b*k17/a+c*k17/a == 0, -b*k16/a+c*k16/a+k4 == 0, -b*k13/a+c*k13/a-b*k21/a+c*k21/a+b*k5/a-c*k5/a == 0, b*k44/a-c*k44/a == 0, -b*k45/a+c*k45/a == 0, -b*k20/a+c*k20/a == 0, -b*k44/a+c*k44/a == 0, b*k46/a-c*k46/a == 0, b^2*k47/a^2-2*b*c*k47/a^2+c^2*k47/a^2 == 0, k3 == 0, -k4 == 0, -b*k12/a+c*k12/a-a*k6/b+c*k6/b == 0, -b*k19/a+c*k19/a+a*k7/c-b*k7/c == 0, b*k45/a-c*k45/a == 0, -b*k46/a+c*k46/a == 0, -k48+c*k48/a+c*k48/b-c^2*k48/(a*b) == 0, -k49+b*k49/a+b*k49/c-b^2*k49/(a*c) == 0, a*k1/b-c*k1/b == 0, a*k4/b-c*k4/b == 0, a*k3/b-c*k3/b+k9 == 0, -k10+a*k2/b-c*k2/b == 0, a*k7/b-c*k7/b == 0, -k9 == 0, k11 == 0, b*k12/a-c*k12/a+a*k6/b-c*k6/b == 0, a*k15/b-c*k15/b == 0, k10+a*k18/b-c*k18/b == 0, -k11+a*k17/b-c*k17/b == 0, a*k16/b-c*k16/b == 0, -a*k13/b+c*k13/b+a*k21/b-c*k21/b+a*k5/b-c*k5/b == 0, -a*k44/b+c*k44/b == 0, a*k45/b-c*k45/b == 0, a*k14/c-b*k14/c+a*k20/b-c*k20/b == 0, a*k44/b-c*k44/b == 0, -a*k46/b+c*k46/b == 0, -k47+c*k47/a+c*k47/b-c^2*k47/(a*b) == 0, a*k19/b-c*k19/b == 0, -a*k45/b+c*k45/b == 0, a*k46/b-c*k46/b == 0, a^2*k48/b^2-2*a*c*k48/b^2+c^2*k48/b^2 == 0, -k49+a*k49/b+a*k49/c-a^2*k49/(b*c) == 0, k16 == 0, -k17 == 0, -a*k1/c+b*k1/c == 0, -k16-a*k4/c+b*k4/c == 0, -a*k3/c+b*k3/c == 0, k18-a*k2/c+b*k2/c == 0, b*k19/a-c*k19/a-a*k7/c+b*k7/c == 0, -a*k6/c+b*k6/c == 0, -a*k8/c+b*k8/c == 0, -a*k11/c+b*k11/c+k17 == 0, -a*k10/c+b*k10/c-k18 == 0, -a*k9/c+b*k9/c == 0, -a*k14/c+b*k14/c-a*k20/b+c*k20/b == 0, -a*k13/c+b*k13/c+a*k21/c-b*k21/c-a*k5/c+b*k5/c == 0, a*k44/c-b*k44/c == 0, -a*k45/c+b*k45/c == 0, -a*k44/c+b*k44/c == 0, a*k46/c-b*k46/c == 0, -k47+b*k47/a+b*k47/c-b^2*k47/(a*c) == 0, -a*k12/c+b*k12/c == 0, a*k45/c-b*k45/c == 0, -a*k46/c+b*k46/c == 0, -k48+a*k48/b+a*k48/c-a^2*k48/(b*c) == 0, a^2*k49/c^2-2*a*b*k49/c^2+b^2*k49/c^2 == 0, k8 == 0, k11 == 0, -k15 == 0, k10-k18 == 0, -k17 == 0, k9 == 0, -k16 == 0, -k29 == 0, k14-k32 == 0, -k21+k23-k31 == 0, -k24-k30 == 0, -k35 == 0, k44 == 0, -k45 == 0, k36 == 0, k13-k23+k39 == 0, -k20+k38 == 0, k25+k37 == 0, b*k26/a-c*k26/a-k34+k42 ==\ > 0, -2*k44 == 0, k45 == 0, k46 == 0, b*k47/a-c*k47/a == 0, k41 == 0, k44 == 0, -k46 == 0, -b*k47/a+c*k47/a == 0, k12+k24 == 0, -k19-k25 == 0, -a*k27/b+c*k27/b-k33 == 0, k45 == 0, -k46 == 0, -a*k48/b+c*k48/b == 0, a*k28/c-b*k28/c+k40 == 0, -k45 == 0, k46 == 0, a*k48/b-c*k48/b == 0, a*k49/c-b*k49/c == 0, -a*k49/c+b*k49/c == 0, -k1 == 0, -k4 == 0, -k3 == 0, k15 == 0, k18-k2 == 0, k17 == 0, k16 == 0, k22 == 0, k25-k7 == 0, k24+k30 == 0, k21+k23-k31 == 0, k28 == 0, -k44 == 0, k45 == 0, -k30-k6 ==\ > 0, k20+k32 == 0, k27+b*k33/a-c*k33/a == 0, k44 == 0, -k46 == 0, -b*k47/a+c*k47/a == 0, -k36 == 0, k31-k39-k5 == 0, -k32-k38 == 0, k19-k37 == 0, k26-a*k34/b+c*k34/b-k42 == 0, k44 == 0, -2*k45 == 0, k46 ==\ > 0, a*k48/b-c*k48/b == 0, a*k35/c-b*k35/c-k41 == 0, -k44 == 0, k46 == 0, b*k47/a-c*k47/a == 0, -a*k49/c+b*k49/c == 0, -k40 == 0, k45 == 0, -k46 ==\ > 0, -a*k48/b+c*k48/b == 0, a*k49/c-b*k49/c == 0, k1 == 0, k4 == 0, k3 == 0, -k8 == 0, -k11 == 0, -k10+k2 == 0, -k9 == 0, k37+k7 == 0, -k14-k38 == 0, -k22 == 0, -k25-k37 == 0, -k24+k6 == 0, -k13-k23+k39 == 0, -k28+b*k40/a-c*k40/a == 0, k44 == 0, -k45 == 0, -k27 == 0, -k44 == 0, k46 == 0, b*k47/a-c*k47/a == 0, k29 == 0, k32+k38 == 0, k31-k39+k5 == 0, -k12+k30 == 0, k35-a*k41/b+c*k41/b == 0, -k44 == 0, k45 == 0, -k26+k34+a*k42/c-b*k42/c == 0, k44 == 0, k45 == 0, -2*k46 == 0, -b*k47/a+c*k47/a == 0, -a*k48/b+c*k48/b == 0, a*k49/c-b*k49/c == 0, k33 ==\ > 0, -k45 == 0, k46 == 0, a*k48/b-c*k48/b == 0, -a*k49/c+b*k49/c == 0 }; 0.04 Second In[109]:= vars = {k1, k2, k3, k4, k5, k6, k7, k8, k9, k10, k11, k12, k13,\ > k14, k15, k16, k17, k18, k19, k20, k21, k22, k23, k24, k25, k26, k27, k28, k29, k30, k31, k32, k33, k34, k35, k36, k37, k38, k39, k40, k41, k42, k43, k44, k45, k46, k47, k48, k49}; 0. Second In[110]:= Solve[eqns, vars] Solve::svars: Equations may not give solutions for all "solve" variables. General::stop: Further output of Solve::svars will be suppressed during this calculation. 1.31 Second Out[110]= {{k22 -> 0, k29 -> 0, k36 -> 0, k23 -> k39, k24 -> 0, k25 -> 0, a k42 > k30 -> 0, k31 -> k39, k32 -> 0, k37 -> 0, k38 -> 0, k26 -> -----, c b k42 > k27 -> 0, k28 -> 0, k33 -> 0, k34 -> -----, k35 -> 0, k40 -> 0, c > k41 -> 0, k1 -> 0, k15 -> 0, k8 -> 0, k10 -> 0, k11 -> 0, k16 -> 0, > k17 -> 0, k18 -> 0, k2 -> 0, k3 -> 0, k4 -> 0, k9 -> 0, k12 -> 0, > k13 -> 0, k14 -> 0, k19 -> 0, k20 -> 0, k21 -> 0, k5 -> 0, k6 -> 0, > k7 -> 0, k44 -> 0, k45 -> 0, k46 -> 0, k47 -> 0, k48 -> 0, k49 -> 0}} In[111]:= (* Solve a 3 x 3 system of nonlinear equations *) 0. Second In[112]:= eqn1 = x^2*y + 3*y*z - 4 == 0 0. Second 2 Out[112]= -4 + x y + 3 y z == 0 In[113]:= eqn2 = -3*x^2*z + 2*y^2 + 1 == 0 0. Second 2 2 Out[113]= 1 + 2 y - 3 x z == 0 In[114]:= eqn3 = 2*y*z^2 - z^2 - 1 == 0 0. Second 2 2 Out[114]= -1 - z + 2 y z == 0 In[115]:= (* Solving this by hand would be a nightmare *) 0. Second In[116]:= Solve[{eqn1, eqn2, eqn3}, {x, y, z}] $IterationLimit::itlim: Iteration limit of 4096 exceeded. 22.99 Second Out[116]= {{x -> -1, y -> 1, z -> 1}, {x -> 1, y -> 1, z -> 1}, 3/4 > {x -> -((-1) Sqrt[-I + Sqrt[2]]), y -> I Sqrt[2], 1 - I Sqrt[2] 3/4 > z -> -------------}, {x -> (-1) Sqrt[-I + Sqrt[2]], y -> I Sqrt[2], 3 1 - I Sqrt[2] 1/4 > z -> -------------}, {x -> -((-1) Sqrt[I + Sqrt[2]]), 3 1 + I Sqrt[2] > y -> -I Sqrt[2], z -> -------------}, 3 1/4 1 + I Sqrt[2] > {x -> (-1) Sqrt[I + Sqrt[2]], y -> -I Sqrt[2], z -> -------------}, 3 2 3 4 5 > {x -> Hold[-Sqrt[(-19 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 2 3 4 5 > 1] + 48 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 1] 2 2 3 4 5 3 > - 2 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 1] - 2 3 4 5 4 > 48 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 1] - 2 3 4 5 5 > 3 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 1] + 2 3 4 5 6 > 48 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 1] - 2 3 4 5 7 > 18 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 1] ) / 6]]\ 2 3 4 5 > , y -> (-2 - 5 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 1] 2 2 3 4 5 4 > - 21 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 1] + 2 3 4 5 5 > 48 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 1] - 2 3 4 5 6 > 18 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 1] ) / 2, 2 3 4 5 > z -> Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 1]}, > {x -> 1 / Sqrt[6 / 2 3 4 5 > (-19 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 1] + 2 3 4 5 2 > 48 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 1] - 2 3 4 5 3 > 2 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 1] - 2 3 4 5 4 > 48 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 1] - 2 3 4 5 5 > 3 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 1] + 2 3 4 5 6 > 48 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 1] - 2 3 4 5 7 > 18 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 1] )], 2 3 4 5 2 > y -> (-2 - 5 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 1] - 2 3 4 5 4 > 21 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 1] + 2 3 4 5 5 > 48 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 1] - 2 3 4 5 6 > 18 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 1] ) / 2, 2 3 4 5 > z -> Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 1]}, 2 3 4 5 > {x -> -Sqrt[(-19 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 2] + 2 3 4 5 2 > 48 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 2] - 2 3 4 5 3 > 2 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 2] - 2 3 4 5 4 > 48 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 2] - 2 3 4 5 5 > 3 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 2] + 2 3 4 5 6 > 48 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 2] - 2 3 4 5 7 > 18 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 2] ) / 6], 2 3 4 5 2 > y -> (-2 - 5 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 2] - 2 3 4 5 4 > 21 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 2] + 2 3 4 5 5 > 48 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 2] - 2 3 4 5 6 > 18 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 2] ) / 2, 2 3 4 5 > z -> Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 2]}, 2 3 4 5 > {x -> Sqrt[(-19 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 2] + 2 3 4 5 2 > 48 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 2] - 2 3 4 5 3 > 2 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 2] - 2 3 4 5 4 > 48 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 2] - 2 3 4 5 5 > 3 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 2] + 2 3 4 5 6 > 48 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 2] - 2 3 4 5 7 > 18 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 2] ) / 6], 2 3 4 5 2 > y -> (-2 - 5 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 2] - 2 3 4 5 4 > 21 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 2] + 2 3 4 5 5 > 48 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 2] - 2 3 4 5 6 > 18 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 2] ) / 2, 2 3 4 5 > z -> Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 2]}, 2 3 4 5 > {x -> -Sqrt[(-19 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 3] + 2 3 4 5 2 > 48 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 3] - 2 3 4 5 3 > 2 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 3] - 2 3 4 5 4 > 48 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 3] - 2 3 4 5 5 > 3 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 3] + 2 3 4 5 6 > 48 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 3] - 2 3 4 5 7 > 18 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 3] ) / 6], 2 3 4 5 2 > y -> (-2 - 5 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 3] - 2 3 4 5 4 > 21 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 3] + 2 3 4 5 5 > 48 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 3] - 2 3 4 5 6 > 18 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 3] ) / 2, 2 3 4 5 > z -> Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 3]}, 2 3 4 5 > {x -> Sqrt[(-19 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 3] + 2 3 4 5 2 > 48 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 3] - 2 3 4 5 3 > 2 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 3] - 2 3 4 5 4 > 48 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 3] - 2 3 4 5 5 > 3 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 3] + 2 3 4 5 6 > 48 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 3] - 2 3 4 5 7 > 18 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 3] ) / 6], 2 3 4 5 2 > y -> (-2 - 5 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 3] - 2 3 4 5 4 > 21 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 3] + 2 3 4 5 5 > 48 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 3] - 2 3 4 5 6 > 18 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 3] ) / 2, 2 3 4 5 > z -> Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 3]}, 2 3 4 5 > {x -> -Sqrt[(-19 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 4] + 2 3 4 5 2 > 48 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 4] - 2 3 4 5 3 > 2 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 4] - 2 3 4 5 4 > 48 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 4] - 2 3 4 5 5 > 3 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 4] + 2 3 4 5 6 > 48 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 4] - 2 3 4 5 7 > 18 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 4] ) / 6], 2 3 4 5 2 > y -> (-2 - 5 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 4] - 2 3 4 5 4 > 21 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 4] + 2 3 4 5 5 > 48 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 4] - 2 3 4 5 6 > 18 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 4] ) / 2, 2 3 4 5 > z -> Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 4]}, 2 3 4 5 > {x -> Sqrt[(-19 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 4] + 2 3 4 5 2 > 48 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 4] - 2 3 4 5 3 > 2 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 4] - 2 3 4 5 4 > 48 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 4] - 2 3 4 5 5 > 3 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 4] + 2 3 4 5 6 > 48 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 4] - 2 3 4 5 7 > 18 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 4] ) / 6], 2 3 4 5 2 > y -> (-2 - 5 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 4] - 2 3 4 5 4 > 21 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 4] + 2 3 4 5 5 > 48 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 4] - 2 3 4 5 6 > 18 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 4] ) / 2, 2 3 4 5 > z -> Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 4]}, 2 3 4 5 > {x -> -Sqrt[(-19 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 5] + 2 3 4 5 2 > 48 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 5] - 2 3 4 5 3 > 2 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 5] - 2 3 4 5 4 > 48 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 5] - 2 3 4 5 5 > 3 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 5] + 2 3 4 5 6 > 48 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 5] - 2 3 4 5 7 > 18 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 5] ) / 6], 2 3 4 5 2 > y -> (-2 - 5 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 5] - 2 3 4 5 4 > 21 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 5] + 2 3 4 5 5 > 48 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 5] - 2 3 4 5 6 > 18 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 5] ) / 2, 2 3 4 5 > z -> Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 5]}, 2 3 4 5 > {x -> Sqrt[(-19 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 5] + 2 3 4 5 2 > 48 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 5] - 2 3 4 5 3 > 2 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 5] - 2 3 4 5 4 > 48 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 5] - 2 3 4 5 5 > 3 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 5] + 2 3 4 5 6 > 48 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 5] - 2 3 4 5 7 > 18 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 5] ) / 6], 2 3 4 5 2 > y -> (-2 - 5 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 5] - 2 3 4 5 4 > 21 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 5] + 2 3 4 5 5 > 48 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 5] - 2 3 4 5 6 > 18 Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 5] ) / 2, 2 3 4 5 > z -> Root[-1 - 3 #1 - 7 #1 - 9 #1 - 6 #1 + 6 #1 & , 5]}} In[117]:= N[%] 0.03 Second Out[117]= {{x -> -1., y -> 1., z -> 1.}, {x -> 1., y -> 1., z -> 1.}, > {x -> 0.605 - 1.16877 I, y -> 1.41421 I, z -> 0.333333 - 0.471405 I}, > {x -> -0.605 + 1.16877 I, y -> 1.41421 I, z -> 0.333333 - 0.471405 I}, > {x -> -0.605 - 1.16877 I, y -> -1.41421 I, z -> 0.333333 + 0.471405 I}, > {x -> 0.605 + 1.16877 I, y -> -1.41421 I, z -> 0.333333 + 0.471405 I}, > {x -> Hold[-1. Sqrt[0.166667 2 3 4 > (-19. 2.06554 + 48. 2.06554 - 2. 2.06554 - 48. 2.06554 - 5 6 7 > 3. 2.06554 + 48. 2.06554 - 18. 2.06554 )]], y -> 0.617193, > z -> 2.06554}, {x -> 0.533221, y -> 0.617193, z -> 2.06554}, > {x -> -1.801 - 0.757707 I, y -> 1.06903 - 1.48122 I, > z -> -0.46266 - 0.317888 I}, > {x -> 1.801 + 0.757707 I, y -> 1.06903 - 1.48122 I, > z -> -0.46266 - 0.317888 I}, > {x -> -1.801 + 0.757707 I, y -> 1.06903 + 1.48122 I, > z -> -0.46266 + 0.317888 I}, > {x -> 1.801 - 0.757707 I, y -> 1.06903 + 1.48122 I, > z -> -0.46266 + 0.317888 I}, > {x -> -0.735491 - 1.68896 I, y -> -1.37763 - 0.535859 I, > z -> -0.0701124 - 0.501152 I}, > {x -> 0.735491 + 1.68896 I, y -> -1.37763 - 0.535859 I, > z -> -0.0701124 - 0.501152 I}, > {x -> -0.735491 + 1.68896 I, y -> -1.37763 + 0.535859 I, > z -> -0.0701124 + 0.501152 I}, > {x -> 0.735491 - 1.68896 I, y -> -1.37763 + 0.535859 I, > z -> -0.0701124 + 0.501152 I}} In[118]:= Clear[eqn1, eqn2, eqn3] 0. Second In[119]:= (* ---------- Quit ---------- *) 0. Second In[120]:= Quit[] real 52.65 user 41.94 sys 0.62