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The system + =, + = has exactly one solution: x = 1, y = 2 The nonlinear system + =, + = has the two solutions (x, y) = (1, 0) and (x, y) = (0, 1), while + + =, + + =, + + = has an infinite number of solutions because the third equation is the first equation plus twice the second one and hence contains no independent information; thus any value of z can be chosen and values of x and y can be ...
The equations 3x + 2y = 6 and 3x + 2y = 12 are independent, because any constant times one of them fails to produce the other one. An independent equation is an equation in a system of simultaneous equations which cannot be derived algebraically from the other equations. [1] The concept typically arises in the context of linear equations.
The number of independent equations in the original system is the number of non-zero rows in the echelon form. The system is inconsistent (no solution) if and only if the last non-zero row in echelon form has only one non-zero entry that is in the last column (giving an equation 0 = c where c is a non-zero constant).
The assumption that ZF is consistent is harmless because adding another axiom to an already inconsistent system cannot make the situation worse. Because of independence, the decision whether to use the axiom of choice (or its negation) in a proof cannot be made by appeal to other axioms of set theory.
In linear systems, indeterminacy occurs if and only if the number of independent equations (the rank of the augmented matrix of the system) is less than the number of unknowns and is the same as the rank of the coefficient matrix. For if there are at least as many independent equations as unknowns, that will eliminate any stretches of overlap ...
It is inconsistent if and only if 0 = 1 is a linear combination (with polynomial coefficients) of the equations (this is Hilbert's Nullstellensatz). If an underdetermined system of t equations in n variables (t < n) has solutions, then the set of all complex solutions is an algebraic set of dimension at least n - t.
The question of whether this ideal is the sum of two properly smaller ideals is independent of ZFC, as was proved by Andreas Blass and Saharon Shelah in 1987. [22] Charles Akemann and Nik Weaver showed in 2003 that the statement "there exists a counterexample to Naimark's problem which is generated by ℵ 1, elements" is independent of ZFC.
In mathematics, a function is a rule for taking an input (in the simplest case, a number or set of numbers) [5] and providing an output (which may also be a number). [5] A symbol that stands for an arbitrary input is called an independent variable, while a symbol that stands for an arbitrary output is called a dependent variable. [6]
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