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If an equation is independent of the other equations in its system, then it provides information beyond that which is provided by the other 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
225:(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.
210:. If it is possible to duplicate one of the equations in a system by multiplying each of the other equations by some number (potentially a different number for each equation) and summing the resulting equations, then that equation is
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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
248:(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.
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on the others. But if this is not possible, then that equation is independent of the others.
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