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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 general, a system with fewer equations than unknowns has infinitely many solutions, but it may have no solution. Such a system is known as an underdetermined system. In general, a system with the same number of equations and unknowns has a single unique solution. In general, a system with more equations than unknowns has no solution.
There will be an infinitude of other solutions only when the system of equations has enough dependencies (linearly dependent equations) that the number of independent equations is at most N − 1. But with M ≥ N the number of independent equations could be as high as N , in which case the trivial solution is the only one.
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 ...
An indeterminate system by definition is consistent, in the sense of having at least one solution. [3] For a system of linear equations, the number of equations in an indeterminate system could be the same as the number of unknowns, less than the number of unknowns (an underdetermined system), or greater than the number of unknowns (an ...
A system is inconsistent if it has no complex solution (or, if the coefficients are not complex numbers, no solution in an algebraically closed field containing the coefficients). By Hilbert's Nullstellensatz this means that 1 is a linear combination (with polynomials as coefficients) of the first members of the equations. Most but not all ...
In mathematics, a set of simultaneous equations, also known as a system of equations or an equation system, is a finite set of equations for which common solutions are sought. An equation system is usually classified in the same manner as single equations, namely as a: System of linear equations, System of nonlinear equations,
Consider the system of equations x + y + 2z = 3, x + y + z = 1, 2x + 2y + 2z = 2.. The coefficient matrix is = [], and the augmented matrix is (|) = [].Since both of these have the same rank, namely 2, there exists at least one solution; and since their rank is less than the number of unknowns, the latter being 3, there are infinitely many solutions.