Independent equation

[1] The concept typically arises in the context of linear equations.

In contrast, if an equation is dependent on the others, then it provides no information not contained in the others collectively, and the equation can be dropped from the system without any information loss.

The number of independent equations in a system of consistent equations (a system that has at least one solution) can never be greater than the number of unknowns.

Equivalently, if a system has more independent equations than unknowns, it is inconsistent and has no solutions.

The concepts of dependence and independence of systems are partially generalized in numerical linear algebra by the condition number, which (roughly) measures how close a system of equations is to being dependent (a condition number of infinity is a dependent system, and a system of orthogonal equations is maximally independent and has a condition number close to 1.)

The equations x − 2 y = −1 , 3 x + 5 y = 8 , and 4 x + 3 y = 7 are linearly dependent, because 1 times the first equation plus 1 times the second equation reproduces the third equation. But any two of them are independent of each other, since any constant times one of them fails to reproduce the other.
The equations 3 x + 2 y = 6 and 3 x + 2 y = 12 are independent, because any constant times one of them fails to produce the other one.
(The image is inaccurate) These equations are linearly dependent because -7 times x+1 plus -5 times -2x-1 delivers (-3x+2)/12. In order to get a solution that is not null, there can be no more than two independent linear equations in a 2D plane.