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How to solve 3 simultaneous algebraic equations with a equality constraint.

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Yokuna
Yokuna on 22 Jun 2021
Commented: Walter Roberson on 27 Jul 2021
If someone could help me to plot x1 vs t from the information. Please if someone could give any idea.
%Initial conditions
x1=140; x2=140; x3=140;
%Equations
x1 =t*x1+x2+t*x3;
x2 = 2*t*x1+t*x2+x3;
x3 = t*x1+x2+x3;
% Equality constraint
x1+x2+x3=420;
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Walter Roberson
Walter Roberson on 27 Jul 2021
That does not appear to be related? please open a new question, and when you do please be more clear what you are asking for.

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Accepted Answer

Walter Roberson
Walter Roberson on 23 Jun 2021
Ran in:
You cannot usefully plot x1 vs t. Your system defines three specific sets of points, two of which are complex-valued
syms x1 x2 x3 t
eqn = [x1 == t*x1+x2+t*x3;
x2 == 2*t*x1+t*x2+x3;
x3 == t*x1+x2+x3;
x1+x2+x3==420]
eqn = 
sol = solve(eqn,[x1, x2, x3, t], 'maxdegree', 3)
sol = struct with fields:
x1: [3×1 sym] x2: [3×1 sym] x3: [3×1 sym] t: [3×1 sym]
[sol.x1, sol.x2, sol.x3, sol.t]
ans = 
vpa(ans,10)
ans = 
so the only real-valued solution is x1 about -235, x2 about 732, x3 about -76, and t about 3.1 .
You might perhaps be expecting all-positive results, but look at your equations:
x3 = t*x1+x2+x3;
x3 appears with coefficient 1 on both sides, so you can subtract it from both sides, leading to
0 == t*x1 + x2
and if t and x1 and x2 are all positive, then that equation cannot be satisfied. It can potentially be satisfied if t and x2 are both 0
If you substitute t = 0 into your first three equations, you can come out with a consistent solution only if x1 = x2 = x3 = 0. However, that does not satisfied the constraint. This establishes that there is no consistent solution for arbitrary times.
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More Answers (1)

ILDEBERTO DE LOS SANTOS RUIZ
You only need to express in terms of t and plot that relationship:
syms x1 x2 x3 t
x3 = solve(x1+x2+x3 == 420,x3)
EQ1 = x1 == t*x1+x2+t*x3;
EQ2 = x2 == 2*t*x1+t*x2+x3;
[x1,x2] = solve(EQ1,EQ2)
ezplot(x1,[0,2])
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Walter Roberson
Walter Roberson on 21 Jul 2021
Ran in:
And the information given includes three equations plus one constraint equation.
syms x1 x2 x3 t
x3 = solve(x1+x2+x3 == 420,x3)
x3 = 
EQ1 = x1 == t*x1+x2+t*x3;
EQ2 = x2 == 2*t*x1+t*x2+x3;
EQ3 = x3 == t*x1+x2+x3;
[x1_12,x2_12] = solve(EQ1,EQ2)
x1_12 = 
x2_12 = 
[x1_13,x2_13] = solve(EQ1,EQ3)
x1_13 = 
x2_13 = 
[x1_23,x2_23] = solve(EQ2,EQ3)
x1_23 = 
x2_23 = 
ezplot(x1_12,[-2,5])
hold on
ezplot(x1_13,[-2 5])
ezplot(x1_23,[-2 5])
hold off
legend({'EQ1,EQ2', 'EQ1,EQ3', 'EQ2,EQ3'}, 'location', 'southwest');
Three very different pairwise answers. It looks like there might be a common answer near t = 3; let us see:
ezplot(x1_12,[2.5,3.5])
hold on
ezplot(x1_13,[2.5,3.5])
ezplot(x1_23,[2.5,3.5])
hold off
legend({'EQ1,EQ2', 'EQ1,EQ3', 'EQ2,EQ3'}, 'location', 'southwest');
tsol = vpasolve(x1_12 == x1_13, 3)
tsol = 
X1 = subs([x1_12, x1_13, x1_23], t, tsol(2))
X1 = 
X2 = subs([x2_12, x2_13, x2_23], t, tsol(2))
X2 = 
So far, so good, the pairs of equation seem to check out.
X3 = subs(x3, [x1, x2], [X1(1), X2(1)])
X3 = 
... which is the solution from the first row of solutions I posted in https://www.mathworks.com/matlabcentral/answers/862790-how-to-solve-3-simultaneous-algebraic-equations-with-a-equality-constraint#answer_731385
That is, there is only one real-valued solution to all of the equations simultaneously.

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