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Common to the class is the nature of the resulting equation, which is a linear Diophantine equation in two unknowns. Most members of the class are determinate, but some are not (the monkey and the coconuts is one of the latter). Familiar algebraic methods are unavailing for solving such equations.
The methods for solving equations generally depend on the type of equation, both the kind of expressions in the equation and the kind of values that may be assumed by the unknowns. The variety in types of equations is large, and so are the corresponding methods.
This does not reduce the generality, as this can be realized by subtracting the right-hand side from both sides. The most common type of equation is a polynomial equation (commonly called also an algebraic equation) in which the two sides are polynomials. The sides of a polynomial equation contain one or more terms. For example, the equation
In linear algebra, Cramer's rule is an explicit formula for the solution of a system of linear equations with as many equations as unknowns, valid whenever the system has a unique solution. It expresses the solution in terms of the determinants of the (square) coefficient matrix and of matrices obtained from it by replacing one column by the ...
Illustrations in Jade Mirror of the Four Unknowns Jia Xian triangle. Jade Mirror of the Four Unknowns, [1] Siyuan yujian (simplified Chinese: 四元玉鉴; traditional Chinese: 四元玉鑒), also referred to as Jade Mirror of the Four Origins, [2] is a 1303 mathematical monograph by Yuan dynasty mathematician Zhu Shijie. [3]
Hilbert's tenth problem is the tenth on the list of mathematical problems that the German mathematician David Hilbert posed in 1900. It is the challenge to provide a general algorithm that, for any given Diophantine equation (a polynomial equation with integer coefficients and a finite number of unknowns), can decide whether the equation has a solution with all unknowns taking integer values.
The squares modulo 4 are congruent to 0 and 1. Thus the left-hand side of the equation is congruent to 0, 1, or 2, and the right-hand side is congruent to 0 or 3. Thus the equality may be obtained only if x, y, and z are all even, and are thus not coprime. Thus the only solution is the trivial solution (0, 0, 0).
In mathematics, a linear equation is an equation that may be put in the form + … + + =, where , …, are the variables (or unknowns), and ,, …, are the coefficients, which are often real numbers. The coefficients may be considered as parameters of the equation and may be arbitrary expressions , provided they do not contain any of the variables.