How to Solve sytem of 7 equations with 6 unknown variables? How to combine any two equations? equations and variables given belowRequest for help

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syms eqn1 eqn2 eqn3 eqn4 eqn5 eqn6 eqn7 eqns sol p1sol p2sol p3sol p4sol p5sol p6sol p1 p2 p3 p4 p5 p6

eqn1 = -p1*0.00074+p2*30+ p5*2 + p6*0.2 == 0

eqn2 = p1*0.00062 - p2*30.00077 == 0;

eqn3 = p1*0.00012 +0.00031*p2 - p3*0.00012 == 0;

eqn4 = p2*0.00046 +p3*0.00012- p4*0.00063 == 0;

eqn5 = p4*0.0004 - p5*2 == 0;

eqn6 = p4*0.00023 - p6*0.2 == 0;

eqn7 = p1+p2+p3+p4+p5+p6==1;

% p1 to p6 above are 6 varaibles with 7 equations

% ? eqns = [eqn1, eqn2, eqn3, eqn4, eqn5, eqn7]

% ? sol = solve([eqns],[p1,p2,p3,p4,p5,p6]);

Answer 1

[David Goodmanson]

Hello Vijaya,

This is an overdetermined set of linear equations of the form M*x = b and can be solved in the least squares sense with the backslash function. The matrix M is (if I have not made a mistake)

M = [-0.00074  30         0        0       2   0.2 
      0.00062 -30.00077   0        0       0   0  
      0.00012   0.00031  -0.00012  0       0   0 
      0         0.00046   0.00012 -0.00063 0   0
      0         0         0        0.0004 -2   0 
      0         0         0        0.00023 0  -0.2  
      1         1         1        1       1   1   ]

cond(M)

     ans = 2.2943e+05

The condition number is large but not unreasonable. Then

b = [0 0 0 0 0 0 1]';
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x = M\b
  
  x =
      0.45645
    9.433e-06
      0.45647
     0.086954
   1.7391e-05
   9.9997e-05

concistency_check =  M*x    % compare with b

concistency_check  =

   1.2769e-17
   7.3885e-18
  -3.6135e-17
   1.0916e-18
   1.3783e-17
    1.119e-18
            1

This result is not so bad.

[Vijaya Kumar Kondagunta]

I am a beginner of Matlab. Grateful to you for the help