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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,
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 system of equations whose left-hand sides are linearly independent is always consistent. Putting it another way, according to the Rouché–Capelli theorem, any system of equations (overdetermined or otherwise) is inconsistent if the rank of the augmented matrix is greater than the rank of the coefficient matrix. If, on the other hand, the ...
Another way of solving the same system of linear equations is by substitution. {+ = = An equivalent for y can be deduced by using one of the two equations. Using the second equation: = Subtracting from each side of the equation:
Putting it another way, according to the Rouché–Capelli theorem, any system of equations (overdetermined or otherwise) is inconsistent if the rank of the augmented matrix is greater than the rank of the coefficient matrix. If, on the other hand, the ranks of these two matrices are equal, the system must have at least one solution.
Two equations or two systems of equations are equivalent, if they have the same set of solutions. The following operations transform an equation or a system of equations into an equivalent one – provided that the operations are meaningful for the expressions they are applied to: Adding or subtracting the same quantity to both sides of an ...
For example, if a system contains , a system over the rational numbers is obtained by adding the equation r 2 2 – 2 = 0 and replacing by r 2 in the other equations. In the case of a finite field, the same transformation allows always supposing that the field k has a prime order.
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 ...