# solving 4 equations with 4 unknowns

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Hamza Harbi on 7 Mar 2023
Edited: John D'Errico on 7 Mar 2023
why do matlab gives two values for each unknown in the following system?
syms a b d e
eqn1 = a+b-4 == 0;
eqn2 = d+e-5 == 0;
eqn3 = a + b/2 + d/2 - 6.5/1.3 == 0;
eqn4 = d*b - exp(1.51)*e*a == 0;
[a,b,d,e]=solve(eqn1,eqn2,eqn3,eqn4)
a = b = d = e = John D'Errico on 7 Mar 2023
Edited: John D'Errico on 7 Mar 2023
Because there are two solutions.
For example, how many solutions would you expect to see for the problem
syms x
xsol = solve(x^2 + x + 2,x)
xsol = Two, of course, this is just a quadratic polynomial. Would you have expected only one solution? If then, which one?
And, while you may not think of your problem as being implicitly a quadratic in a sense it is one. That is, look at the products of variables in eqn4. If you eliminate some of the variables using substitution, at the end, you would find one quadratic polynomial in one unknown. Ergo, two solutions. And once you then have two solutions for that final variable, substituting back in to recover the other variables yields EXACTLY two solutions for each of them.

William Rose on 7 Mar 2023
Edited: William Rose on 7 Mar 2023
@Hamza Harbi, because the equation has two solutions. The first values of a,b,d,e are one solution; the second values of a,b,d,e are another.