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1. Existence and Uniqueness Theorem. The system Ax = b has a solution if and only if rank (A) = rank(A, b). The solution is unique if and only if A is invertible.

A system of linear equations is said to be consistent if there is one solution that satisfies all of the equations. The solution to both are the line x + y = 10, thus the system is consistent.

In mathematics, an existence theorem is a theorem which asserts the existence of a certain object. It might be a statement which begins with the phrase "there exist(s)", or it might be a universal statement whose last quantifier is existential (e.g., "for all x, y, ... there exist(s) ...").

Picard's Existence and Uniqueness Theorem Consider the Initial Value Problem (IVP) y0 = f(x, y), y(x0) = y0. produces a sequence of functions {yn(x)} that converges to this solution uniformly on I.

(i) If the lines intersect at a point, then that point gives the unique solution of the two equations. In this case, the pair of equations is consistent. (ii) If the lines coincide, then there are infinitely many solutions ? each point on the line being a solution.

Existence and Uniqueness Theorem. The existence and uniqueness theorem for initial value problems of ordinary differential equations implies the condition for the existence of a solution of linear or non-linear initial value problems and ensures the uniqueness of the obtained solution.

A system is said to be consistent if it only has one solution. A consistent system is independent if it has only one solution. Consider the following equations: x ? y = 2 and x + y = 6. Equation x + y = 6 has multiple solutions, but both equations have one solution.

Existence and Uniqueness Theorem. The system Ax = b has a solution if and only if rank (A) = rank(A, b). The solution is unique if and only if A is invertible.

An existence and uniqueness theorem, asserting that by the prescription of some freely chosen functions, one can single out one specific solution of the PDE. by continuously changing the free choices, one continuously changes the corresponding solution.