11_8_42_Solving systems of equations

397 days ago by Cygnus_X1

# Solving the problem as given, i.e. with TWO constraints var('x y z p q') eq1 = 2*(p+q)*x==1 eq2 = -2*p*y ==1 eq3 = -p+2*q*z ==1 eq4 = x^2 - y^2 ==z eq5 = x^2+z^2 == 4 solve([eq1, eq2,eq3,eq4,eq5], x, y, z, p, q) 
        
[[x == -1.65287821811, y == -1.96419410004, z == -1.1260523322, p ==
0.254557327094, q == -0.557059961315], [x == 1.89517799034, y ==
1.71834681828, z == 0.638983878847, p == -0.29097733908, q ==
0.554804804805], [x == (0.692503443362 + 0.69105161484*I), y ==
(-0.405011339054 - 1.47499099146*I), z == (2.01357292481 -
0.237664907449*I), p == (0.0865544738488 - 0.315218505969*I), q ==
(0.275212207638 - 0.0457897342537*I)], [x == (0.692503443362 -
0.69105161484*I), y == (-0.405011339054 + 1.47499099146*I), z ==
(2.01357292481 + 0.237664907449*I), p == (0.0865544738488 +
0.315218505969*I), q == (0.275212207638 + 0.0457897342537*I)], [x ==
(0.434002834389 - 1.6352260808*I), y == (-0.781339489634 +
0.731018210543*I), z == (-2.56170964898 - 0.277038716787*I), p ==
(0.341232337425 + 0.319255657743*I), q == (-0.265419288831 -
0.0336089649028*I)], [x == (0.434002834389 + 1.6352260808*I), y ==
(-0.781339489634 - 0.731018210543*I), z == (-2.56170964898 +
0.277038716787*I), p == (0.341232337425 - 0.319255657743*I), q ==
(-0.265419288831 + 0.0336089649028*I)], [x == -0.992512479201, y ==
1.64967668315, z == -1.73635235732, p == -0.303089675961, q ==
-0.200682319888], [x == -1.5028, y == 0.968872257187, z ==
1.31969407266, p == -0.516063888379, q == 0.183351618212]]
[[x == -1.65287821811, y == -1.96419410004, z == -1.1260523322, p == 0.254557327094, q == -0.557059961315], [x == 1.89517799034, y == 1.71834681828, z == 0.638983878847, p == -0.29097733908, q == 0.554804804805], [x == (0.692503443362 + 0.69105161484*I), y == (-0.405011339054 - 1.47499099146*I), z == (2.01357292481 - 0.237664907449*I), p == (0.0865544738488 - 0.315218505969*I), q == (0.275212207638 - 0.0457897342537*I)], [x == (0.692503443362 - 0.69105161484*I), y == (-0.405011339054 + 1.47499099146*I), z == (2.01357292481 + 0.237664907449*I), p == (0.0865544738488 + 0.315218505969*I), q == (0.275212207638 + 0.0457897342537*I)], [x == (0.434002834389 - 1.6352260808*I), y == (-0.781339489634 + 0.731018210543*I), z == (-2.56170964898 - 0.277038716787*I), p == (0.341232337425 + 0.319255657743*I), q == (-0.265419288831 - 0.0336089649028*I)], [x == (0.434002834389 + 1.6352260808*I), y == (-0.781339489634 - 0.731018210543*I), z == (-2.56170964898 + 0.277038716787*I), p == (0.341232337425 - 0.319255657743*I), q == (-0.265419288831 + 0.0336089649028*I)], [x == -0.992512479201, y == 1.64967668315, z == -1.73635235732, p == -0.303089675961, q == -0.200682319888], [x == -1.5028, y == 0.968872257187, z == 1.31969407266, p == -0.516063888379, q == 0.183351618212]]
sqrt(-1) 
        
I
I
# Seeing that from the equation of the first constraint, one can solve for z, you get to eliminate both z and q. # So now you could solve an equivalent problem with two less variables, no z, and only one constraint. var('x y p') eq1 = 2*x+1 == p*x*(4*x^2-4*y^2+2) eq2 = -2*y+1 == 4*p*y*(y^2-x^2) eq3 = x^4 +(1-2*y^2)*x^2+y^4 == 4 solve([eq1, eq2, eq3], x, y, p) 
        
[[x == -1.65287821811, y == -1.96419410004, p == -0.557059961315], [x ==
1.89517799034, y == 1.71834681828, p == 0.554804804805], [x ==
(0.692503443362 + 0.69105161484*I), y == (-0.405011339054 -
1.47499099146*I), p == (0.275212207638 - 0.0457897342537*I)], [x ==
(0.692503443362 - 0.69105161484*I), y == (-0.405011339054 +
1.47499099146*I), p == (0.275212207638 + 0.0457897342537*I)], [x ==
(0.434002834389 - 1.6352260808*I), y == (-0.781339489634 +
0.731018210543*I), p == (-0.265419288831 - 0.0336089649028*I)], [x ==
(0.434002834389 + 1.6352260808*I), y == (-0.781339489634 -
0.731018210543*I), p == (-0.265419288831 + 0.0336089649028*I)], [x ==
-0.992512479201, y == 1.64967668315, p == -0.200682319888], [x ==
-1.5028, y == 0.968872257187, p == 0.183351618212]]
[[x == -1.65287821811, y == -1.96419410004, p == -0.557059961315], [x == 1.89517799034, y == 1.71834681828, p == 0.554804804805], [x == (0.692503443362 + 0.69105161484*I), y == (-0.405011339054 - 1.47499099146*I), p == (0.275212207638 - 0.0457897342537*I)], [x == (0.692503443362 - 0.69105161484*I), y == (-0.405011339054 + 1.47499099146*I), p == (0.275212207638 + 0.0457897342537*I)], [x == (0.434002834389 - 1.6352260808*I), y == (-0.781339489634 + 0.731018210543*I), p == (-0.265419288831 - 0.0336089649028*I)], [x == (0.434002834389 + 1.6352260808*I), y == (-0.781339489634 - 0.731018210543*I), p == (-0.265419288831 + 0.0336089649028*I)], [x == -0.992512479201, y == 1.64967668315, p == -0.200682319888], [x == -1.5028, y == 0.968872257187, p == 0.183351618212]]