# Solving a system with 4 variables and 4 equations

I am trying to solve a linear system of equations with Mathematica. My variables are x1, x2, x3, x4. I use the code shown below, but it gives error saying that:

Equations may not give solutions for all "solve" variables.

Even when I used LinearSolve, I got the same error.

Solve[{(x1 - ((-m3* n2 + m2 * z3 - 0.5*m2*z3)* x2 + (0.5 *n2* z3)*
x3 + (0.5 *m2 * z3)*x4)/((-1.5) n2* z3)) == (2/
1.5)   ,  (x2 -  ((m3* n2 - 0.5*m3 *n2 - m2 * z3)*
x1 + (0.5* m3* n2)* x3 + (0.5* m2* m3)*
x4)/((-1.5) (m2* m3)) ) == (-m3* n2 + m2* z3 )/((-1.5) m2*
m3)  ,    (x3 - ((-n2*  z1 + .5*n2 * z1)*
x1 + (.5* m2 * z1 - m2 * z1)*x2 + (m1 *n2  - 0.5* m2* z1)*
x4 )/((1.5) n2 z1) ) ==  (2/1.5 )  ,   (
x4 - ((-m1 *n2 + 0.5*m1 *n2 )*x1 + (-0.5 *m1* m2) *
x2 + (-.5* m1* n2 + m2 *z1)*x3)/((1.5) m1* m2) ) == (
m1* n2 - m2* z1)/((1.5)* m1* m2)} , {x1, x2, x3, x4}]


• Apparently, your equations have a certain redundancy such that there is a 1-paremeterfamily of solutions (parameterized by x1). – Henrik Schumacher Sep 7 '18 at 22:46
• You can select which of your variables is the parameter: FullSimplify[ Solve[eqns//Rationalize, #, Reals] & /@ Subsets[{x1, x2, x3, x4}, {3}]] – Bob Hanlon Sep 7 '18 at 23:56
• Use Reduce instead of Solve. – user64494 Sep 8 '18 at 4:42
• In this case, you probably want to use 1/2 rather than 0.5. This uses exact numbers rather than approximate numbers. – mikado Sep 8 '18 at 8:00
• @BobHanlon , I still get the same problem. – mah Sep 10 '18 at 20:16

Extended comment.

eq = Rationalize@{(x1 - ((-m3*n2 + m2*z3 - 0.5*m2*z3)*x2 + (0.5*n2*z3)*
x3 + (0.5*m2*z3)*x4)/((-1.5) n2*z3)) == (2/
1.5), (x2 - ((m3*n2 - 0.5*m3*n2 - m2*z3)*x1 + (0.5*m3*n2)*
x3 + (0.5*m2*m3)*x4)/((-1.5) (m2*m3))) == (-m3*n2 +
m2*z3)/((-1.5) m2*
m3), (x3 - ((-n2*z1 + .5*n2*z1)*x1 + (.5*m2*z1 - m2*z1)*
x2 + (m1*n2 - 0.5*m2*z1)*x4)/((1.5) n2 z1)) == (2/
1.5), (x4 - ((-m1*n2 + 0.5*m1*n2)*x1 + (-0.5*m1*m2)*
x2 + (-.5*m1*n2 + m2*z1)*x3)/((1.5) m1*m2)) == (m1*n2 -
m2*z1)/((1.5)*m1*m2)};
sol = Solve[eq, {x1, x2, x3, x4}][];


The generic solution looks like this:

{x1, x2, x3, x4} /. sol


{x1, -(z3/m3) + (x1 z3)/ m3, -((m1 x1 (m3 n2 + m2 z3))/(m3 (m1 n2 + m2 z1))) - (-2 m1 m3 n2 - m2 m3 z1 - m1 m2 z3)/(m3 (m1 n2 + m2 z1)), (z1 (m3 n2 + m2 z3))/( m3 (m1 n2 + m2 z1)) - (x1 z1 (m3 n2 + m2 z3))/(m3 (m1 n2 + m2 z1))}

Let's check that these really are solutions by substituting the solutions rules sol into the equation eq:

eq /. sol // Simplify


{True, True, True, True}

So the equations do not specify a single solution. Because there are 4 equations and four variables, that means that the equations are somewhat redundant. Usually, one would expect to get only finitely many (or countably many) solution. This is not the case here: We have a continuum of solutions. Because that is somewhat unexpected (and because this often causes problems), Solve warns us. That's all. The message is a warning, not an error.

• I got it. Your explanation was great. – mah Sep 10 '18 at 20:44
• If you like, you can mark it as "accepted", signaling that your question is answered. – Henrik Schumacher Sep 10 '18 at 20:53