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I have a bunch of equations in the form as follows.

a+b+c+d=10
c+d+e+f=12
d+e+c=13

Where I am tying to calculate the values of each variable (many more equations in the actual list ofc).

However, I cannot separate the values of a and b (I can only calculate a+b) or d and e (I can only calculate d+e). How would I detect these "incalculable" variables (such that I know a and b must be calculated together and so forth)?

These seems common enough and I know I've seen software do this, but the term for this escapes and and how I would calculate this is well beyond me.

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    $\begingroup$ This is the subject of linear algebra. In modern terms, you are asking how to calculate the kernel (or "null space") of a linear transformation. There are algorithms and they all are versions of solving the system. $\endgroup$ – whuber Nov 28 '20 at 16:40

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