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ax − by = c where c = R 2 − R 1. Find an integer such that its product with a given integer being increased or decreased by another given integer and then divided by a third integer leaves no remainder. Letting the integer to be determined be x and the three integers be a, b and c, the problem is to find x such that (ax ± b)/c is an integer y.
This is a linear Diophantine equation, related to Bézout's identity. + = + The smallest nontrivial solution in positive integers is 12 3 + 1 3 = 9 3 + 10 3 = 1729.It was famously given as an evident property of 1729, a taxicab number (also named Hardy–Ramanujan number) by Ramanujan to Hardy while meeting in 1917. [1]
When there is only one variable, polynomial equations have the form P(x) = 0, where P is a polynomial, and linear equations have the form ax + b = 0, where a and b are parameters. To solve equations from either family, one uses algorithmic or geometric techniques that originate from linear algebra or mathematical analysis.
Both terms in ax + by are divisible by g; therefore, c must also be divisible by g, or the equation has no solutions. By dividing both sides by c/g, the equation can be reduced to Bezout's identity sa + tb = g, where s and t can be found by the extended Euclidean algorithm. [69] This provides one solution to the Diophantine equation, x 1 = s (c ...
In the simple case of a function of one variable, say, h(x), we can solve an equation of the form h(x) = c for some constant c by considering what is known as the inverse function of h. Given a function h : A → B, the inverse function, denoted h −1 and defined as h −1 : B → A, is a function such that
The solutions of the quadratic equation ax 2 + bx + c = 0 correspond to the roots of the function f(x) = ax 2 + bx + c, since they are the values of x for which f(x) = 0. If a, b, and c are real numbers and the domain of f is the set of real numbers, then the roots of f are exactly the x-coordinates of the points where the graph touches the x-axis.
The answer is that these two matrices are similar exactly when there exists a matrix X such that AX − XB = C. In other words, X is a solution to a Sylvester equation. This is known as Roth's removal rule. [4] One easily checks one direction: If AX − XB = C then
A cubic equation can be solved by compass-and-straightedge construction (without trisector) if and only if it has a rational root. This implies that the old problems of angle trisection and doubling the cube , set by ancient Greek mathematicians , cannot be solved by compass-and-straightedge construction.