Gaussian Elimination Type Method Required

Question

I'm struggling a bit with the following problem:

$3 + 14x = 1 + 25y = 9 + 288z$

I have a series of these equations which I need to solve, with different first terms in each case and one of these first terms changes for each equation: e.g. the second such equation is:

$3 + 14x = 1 + 25y = 47 + 288z$

Obviously, I'd prefer not to brute force this problem, or I wouldn't be posting here. But I just thought there might be an analytical solution, because as I progress further with this, the $x$, $y$ and $z$ coefficients will get larger. Also, if you could advise as to a general solution for any number of terms, I'd welcome it.

Thanks for your help!!!

Answer 1

What you show seems to express relations between the variables rather than equations.

Let us take these as $$a+bx=c+dy=e+fz$$ assuming that none of the coefficients $a,b,c,d,e,f$ is equal to $0$.

Using $a+bx=c+dy$, you can express $y$ as a function of $x$; using $a+bx=e+fz$ you can express $z$ as a function of $x$.

Just do it, and, when required, replace the letters by numbers.

Answer 2

The equations are underdetermined, you can solve for $y$ and $z$ in terms of $x$ and such solutions can be written down immediately with no trouble.