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# Help me to solve the simultaneous equations

Posted 10 years ago
 HI I have a simultaneous equation in terms of a, b, c, d. when I try to use NSolve to solve the equation and get the vales of a, b, c, d in terms of p,q,r,s it is just giving empty bracket as an answer. Any one can help me to solve this equation. NSolve[{a*b + a*c + b*d + c*d == p && a*b + b*c - b*d == q && a*b + b*c + a*d + c*d == r && a*b + a*c - a*d == s}, {a, b, c, d}]  Thanks in advance.
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Posted 10 years ago
 That means there is no solution. Also since there are nonnumeric parameters, you might as well use Solve because that's what NSolve will have to do anyway.
Posted 10 years ago
 Possibly solve the equations stepwise. For example In[49]:= Solve[a*b + a*c + b*d + c*d == p, a] Out[49]= {{a -> (-b d - c d + p)/(b + c)}} In[52]:= a = (-b d - c d + p)/(b + c); Then use this result in second equation for b, etc.
Posted 10 years ago
 In[3]:= Reduce[{a*b + a*c + b*d + c*d == p && a*b + b*c - b*d == q && a*b + b*c + a*d + c*d == r && a*b + a*c - a*d == s}, {a, b, c, d}, Backsubstitution -> True] Out[3]= (p == -q + r + s && a != 0 && b == (a^2 + s - Sqrt[a^4 - 4 a^2 q + 2 a^2 s + s^2])/(2 a) && a^2 + s - Sqrt[a^4 - 4 a^2 q + 2 a^2 s + s^2] != 0 && c == (-3 a^4 + 4 a^2 q - 4 a^2 s - s^2 + 3 a^2 Sqrt[a^4 - 4 a^2 q + 2 a^2 s + s^2] + s Sqrt[a^4 - 4 a^2 q + 2 a^2 s + s^2] - Sqrt[2] a^2 \[Sqrt](-(1/( a^4))(a^4 - 4 a^2 q + 4 a^2 r + 2 a^2 s + s^2) (-a^4 + 2 a^2 q - 2 a^2 s - s^2 + a^2 Sqrt[a^4 - 4 a^2 q + 2 a^2 s + s^2] + s Sqrt[a^4 - 4 a^2 q + 2 a^2 s + s^2])))/(2 a (a^2 + s - Sqrt[a^4 - 4 a^2 q + 2 a^2 s + s^2])) && d == -((a^4 + 2 a^2 s + s^2 - a^2 Sqrt[a^4 - 4 a^2 q + 2 a^2 s + s^2] - s Sqrt[a^4 - 4 a^2 q + 2 a^2 s + s^2] + Sqrt[2] a^2 \[Sqrt](-(1/( a^4))(a^4 - 4 a^2 q + 4 a^2 r + 2 a^2 s + s^2) (-a^4 + 2 a^2 q - 2 a^2 s - s^2 + a^2 Sqrt[a^4 - 4 a^2 q + 2 a^2 s + s^2] + s Sqrt[a^4 - 4 a^2 q + 2 a^2 s + s^2])))/(2 a (a^2 + s - Sqrt[a^4 - 4 a^2 q + 2 a^2 s + s^2])))) || (p == -q + r + s && a != 0 && b == (a^2 + s - Sqrt[a^4 - 4 a^2 q + 2 a^2 s + s^2])/(2 a) && a^2 + s - Sqrt[a^4 - 4 a^2 q + 2 a^2 s + s^2] != 0 && c == (-3 a^4 + 4 a^2 q - 4 a^2 s - s^2 + 3 a^2 Sqrt[a^4 - 4 a^2 q + 2 a^2 s + s^2] + s Sqrt[a^4 - 4 a^2 q + 2 a^2 s + s^2] + Sqrt[2] a^2 \[Sqrt](-(1/( a^4))(a^4 - 4 a^2 q + 4 a^2 r + 2 a^2 s + s^2) (-a^4 + 2 a^2 q - 2 a^2 s - s^2 + a^2 Sqrt[a^4 - 4 a^2 q + 2 a^2 s + s^2] + s Sqrt[a^4 - 4 a^2 q + 2 a^2 s + s^2])))/(2 a (a^2 + s - Sqrt[a^4 - 4 a^2 q + 2 a^2 s + s^2])) && d == -((a^4 + 2 a^2 s + s^2 - a^2 Sqrt[a^4 - 4 a^2 q + 2 a^2 s + s^2] - s Sqrt[a^4 - 4 a^2 q + 2 a^2 s + s^2] - Sqrt[2] a^2 \[Sqrt](-(1/( a^4))(a^4 - 4 a^2 q + 4 a^2 r + 2 a^2 s + s^2) (-a^4 + 2 a^2 q - 2 a^2 s - s^2 + a^2 Sqrt[a^4 - 4 a^2 q + 2 a^2 s + s^2] + s Sqrt[a^4 - 4 a^2 q + 2 a^2 s + s^2])))/(2 a (a^2 + s - Sqrt[a^4 - 4 a^2 q + 2 a^2 s + s^2])))) || (p == -q + r + s && a != 0 && b == (a^2 + s + Sqrt[a^4 - 4 a^2 q + 2 a^2 s + s^2])/(2 a) && a^2 + s + Sqrt[a^4 - 4 a^2 q + 2 a^2 s + s^2] != 0 && c == (-3 a^4 + 4 a^2 q - 4 a^2 s - s^2 - 3 a^2 Sqrt[a^4 - 4 a^2 q + 2 a^2 s + s^2] - s Sqrt[a^4 - 4 a^2 q + 2 a^2 s + s^2] - Sqrt[2] a^2 \[Sqrt]((1/( a^4))(a^4 - 4 a^2 q + 4 a^2 r + 2 a^2 s + s^2) (a^4 - 2 a^2 q + 2 a^2 s + s^2 + a^2 Sqrt[a^4 - 4 a^2 q + 2 a^2 s + s^2] + s Sqrt[a^4 - 4 a^2 q + 2 a^2 s + s^2])))/(2 a (a^2 + s + Sqrt[a^4 - 4 a^2 q + 2 a^2 s + s^2])) && d == (-a^4 - 2 a^2 s - s^2 - a^2 Sqrt[a^4 - 4 a^2 q + 2 a^2 s + s^2] - s Sqrt[a^4 - 4 a^2 q + 2 a^2 s + s^2] - Sqrt[2] a^2 \[Sqrt]((1/( a^4))(a^4 - 4 a^2 q + 4 a^2 r + 2 a^2 s + s^2) (a^4 - 2 a^2 q + 2 a^2 s + s^2 + a^2 Sqrt[a^4 - 4 a^2 q + 2 a^2 s + s^2] + s Sqrt[a^4 - 4 a^2 q + 2 a^2 s + s^2])))/(2 a (a^2 + s + Sqrt[a^4 - 4 a^2 q + 2 a^2 s + s^2]))) || (p == -q + r + s && a != 0 && b == (a^2 + s + Sqrt[a^4 - 4 a^2 q + 2 a^2 s + s^2])/(2 a) && a^2 + s + Sqrt[a^4 - 4 a^2 q + 2 a^2 s + s^2] != 0 && c == (-3 a^4 + 4 a^2 q - 4 a^2 s - s^2 - 3 a^2 Sqrt[a^4 - 4 a^2 q + 2 a^2 s + s^2] - s Sqrt[a^4 - 4 a^2 q + 2 a^2 s + s^2] + Sqrt[2] a^2 \[Sqrt]((1/( a^4))(a^4 - 4 a^2 q + 4 a^2 r + 2 a^2 s + s^2) (a^4 - 2 a^2 q + 2 a^2 s + s^2 + a^2 Sqrt[a^4 - 4 a^2 q + 2 a^2 s + s^2] + s Sqrt[a^4 - 4 a^2 q + 2 a^2 s + s^2])))/(2 a (a^2 + s + Sqrt[a^4 - 4 a^2 q + 2 a^2 s + s^2])) && d == (-a^4 - 2 a^2 s - s^2 - a^2 Sqrt[a^4 - 4 a^2 q + 2 a^2 s + s^2] - s Sqrt[a^4 - 4 a^2 q + 2 a^2 s + s^2] + Sqrt[2] a^2 \[Sqrt]((1/( a^4))(a^4 - 4 a^2 q + 4 a^2 r + 2 a^2 s + s^2) (a^4 - 2 a^2 q + 2 a^2 s + s^2 + a^2 Sqrt[a^4 - 4 a^2 q + 2 a^2 s + s^2] + s Sqrt[a^4 - 4 a^2 q + 2 a^2 s + s^2])))/(2 a (a^2 + s + Sqrt[a^4 - 4 a^2 q + 2 a^2 s + s^2]))) || (s == 0 && p == -q + r && a == 0 && b != 0 && c == (-b^2 + q - Sqrt[b^4 - 2 b^2 q + q^2 + 4 b^2 r])/(2 b) && d == (-b^2 - q - Sqrt[b^4 - 2 b^2 q + q^2 + 4 b^2 r])/( 2 b)) || (s == 0 && p == -q + r && a == 0 && b != 0 && c == (-b^2 + q + Sqrt[b^4 - 2 b^2 q + q^2 + 4 b^2 r])/(2 b) && d == (-b^2 - q + Sqrt[b^4 - 2 b^2 q + q^2 + 4 b^2 r])/( 2 b)) || (q == 0 && p == r + s && b == 0 && a != 0 && c == (-a^2 + s - Sqrt[a^4 + 4 a^2 r + 2 a^2 s + s^2])/( 2 a) && -a^2 + s - Sqrt[a^4 + 4 a^2 r + 2 a^2 s + s^2] != 0 && d == -((a (a^2 + 2 r + s + Sqrt[a^4 + 4 a^2 r + 2 a^2 s + s^2]))/( a^2 - s + Sqrt[a^4 + 4 a^2 r + 2 a^2 s + s^2]))) || (q == 0 && p == r + s && b == 0 && a != 0 && c == (-a^2 + s + Sqrt[a^4 + 4 a^2 r + 2 a^2 s + s^2])/( 2 a) && -a^2 + s + Sqrt[a^4 + 4 a^2 r + 2 a^2 s + s^2] != 0 && d == -((a (a^2 + 2 r + s - Sqrt[a^4 + 4 a^2 r + 2 a^2 s + s^2]))/( a^2 - s - Sqrt[a^4 + 4 a^2 r + 2 a^2 s + s^2]))) || (r == -s && q == 0 && p == 0 && b == 0 && c == 0 && a != 0 && d == -(s/a)) || (s == 0 && q == 0 && p == r && a == 0 && b == 0 && c != 0 && d == r/c) || (s == 0 && r == 0 && q == 0 && p == 0 && a == 0 && b == 0 && c == 0) 
Posted 10 years ago
 together