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An underdetermined linear system has either no solution or infinitely many solutions. For example, + + = + + = is an underdetermined system without any solution; any system of equations having no solution is said to be inconsistent. On the other hand, the system
In mathematics, particularly in algebra, an indeterminate system is a system of simultaneous equations (e.g., linear equations) which has more than one solution (sometimes infinitely many solutions). [1] In the case of a linear system, the system may be said to be underspecified, in which case the presence of more than one solution would imply ...
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 ...
A solution of a linear system is an assignment of values to the variables ,, …, such that each of the equations is satisfied. The set of all possible solutions is called the solution set. [5] A linear system may behave in any one of three possible ways: The system has infinitely many solutions.
Also, the output of a linear system can contain harmonics (and have a smaller fundamental frequency than the input) even when the input is a sinusoid. For example, consider a system described by () = (+ ()) (). It is linear because it satisfies the superposition principle.
In general, any infinite series is the limit of its partial sums. For example, an analytic function is the limit of its Taylor series, within its radius of convergence. = =. This is known as the harmonic series. [6]
[1] [citation needed] An overdetermined system is almost always inconsistent (it has no solution) when constructed with random coefficients. However, an overdetermined system will have solutions in some cases, for example if some equation occurs several times in the system, or if some equations are linear combinations of the others.
For example, the equation + = is a simple indeterminate equation, as is =. Indeterminate equations cannot be solved uniquely. In fact, in some cases it might even have infinitely many solutions. [2] Some of the prominent examples of indeterminate equations include: Univariate polynomial equation: