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When the equations are independent, each equation contains new information about the variables, and removing any of the equations increases the size of the solution set. For linear equations, logical independence is the same as linear independence. The equations x − 2y = −1, 3x + 5y = 8, and 4x + 3y = 7 are linearly dependent. For example ...
Each unknown can be seen as an available degree of freedom. Each equation introduced into the system can be viewed as a constraint that restricts one degree of freedom. Therefore, the critical case (between overdetermined and underdetermined) occurs when the number of equations and the number of free variables are equal.
Let the system of equations be written in matrix form as = where is the coefficient matrix, is the vector of unknowns, and is an vector of constants. In which case, if the system is indeterminate, then the infinite solution set is the set of all vectors generated by [4]
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 ...
Any system of linear equations can be written as a matrix equation. The previous system of equations (in Diagram #1) can be written as follows: [] [] = [] Notice that the rows of the coefficient matrix (corresponding to equations) outnumber the columns (corresponding to unknowns), meaning that the system is overdetermined.
[4] [5] [6] Cramer's rule, implemented in a naive way, is computationally inefficient for systems of more than two or three equations. [7] In the case of n equations in n unknowns, it requires computation of n + 1 determinants, while Gaussian elimination produces the result with the same computational complexity as the computation of a single ...
An example of linear Diophantine equation is ax + by = c where a, b, and c are constants. An exponential Diophantine equation is one for which exponents of the terms of the equation can be unknowns. Diophantine problems have fewer equations than unknown variables and involve finding integers that work correctly for all equations.
For example, the equation x + y = 2x – 1 is solved for the unknown x by the expression x = y + 1, because substituting y + 1 for x in the equation results in (y + 1) + y = 2(y + 1) – 1, a true statement. It is also possible to take the variable y to be the unknown, and then the equation is solved by y = x – 1.
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