# Simultaneous equations

hello!

I am really wondering what is going on here

let's say we have 3 equations:
with x, y, z to be our unknowns and the rest regular integers

a*x+b*y-c*z=0
d*x-e*y+f*z=0
g*x+h*y+i*z=0

the signs of each integer may be positive or negative

is there a solution to this system of equations or the only solution is x, y, z = 0 ?

thanks!

nasu
Gold Member
It depends on the values of the coefficients.
If the 3 by 3 determinant of the coefficients is zero there are non-trivial solutions. See "homogeneous system of equation" for more details.

what you mean "If the 3 by 3 determinant of the coefficients is zero there are non-trivial solutions"?
all the three coefficients are normal integers, either positive or negative

PeroK
Homework Helper
Gold Member
2020 Award
what you mean "If the 3 by 3 determinant of the coefficients is zero there are non-trivial solutions"?
all the three coefficients are normal integers, either positive or negative
You may need to learn a little linear algebra to understand solutions to these equations. Basically, if you arrange your integer coefficients into a 3x3 matrix and your variables into a vector (x, y, z), then you have matrix/vector equation.

If the matrix has an inverse, then the only solution is (0, 0, 0).

If, however, the matrix has no inverse, then you will have infinitely many solutions.

PS: A matrix has an inverse iff its determinant is non-zero.

SteamKing
Staff Emeritus
Homework Helper
what you mean "If the 3 by 3 determinant of the coefficients is zero there are non-trivial solutions"?
all the three coefficients are normal integers, either positive or negative
Yes, but the determinant of a matrix is a single special quantity, a single number, which is computed from the 9 individual coefficients a - i and their location in the matrix.

See http://en.wikipedia.org/wiki/Determinant

What you are calling coefficients (apparently x, y, and z) from the context of your reply, are actually the unknowns or the variables.

no, by coefficients I mean a,b,c,...

nasu
Gold Member
Then you have 9 coefficients for the system.
The matrix of the coefficients is
a b c
d e f
g h i

The determinant of this matrix tells you if there are other solutions than the trivial (0,0,0).
See the link given above by SteamKing.

The idea is that if the equations are independent there is no simultaneous solution (but zero).

Last edited:
SteamKing
Staff Emeritus