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Therefore, the solution = is extraneous and not valid, and the original equation has no solution. For this specific example, it could be recognized that (for the value x = − 2 {\displaystyle x=-2} ), the operation of multiplying by ( x − 2 ) ( x + 2 ) {\displaystyle (x-2)(x+2)} would be a multiplication by zero.
Homogeneous Diophantine equations of degree two are easier to solve. The standard solving method proceeds in two steps. One has first to find one solution, or to prove that there is no solution. When a solution has been found, all solutions are then deduced. For proving that there is no solution, one may reduce the equation modulo p. For ...
Equivalent statement 2: x n + y n = z n, where integer n ≥ 3, has no non-trivial solutions x, y, z ∈ Q. This is because the exponents of x, y, and z are equal (to n), so if there is a solution in Q, then it can be multiplied through by an appropriate common denominator to get a solution in Z, and hence in N.
One particular solution is x = 0, y = 0, z = 0. Two other solutions are x = 3, y = 6, z = 1, and x = 8, y = 9, z = 2. There is a unique plane in three-dimensional space which passes through the three points with these coordinates, and this plane is the set of all points whose coordinates are solutions of the equation.
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 ...
In mathematics, the solution set of a system of equations or inequality is the set of all its solutions, that is the values that satisfy all equations and inequalities. [1] Also, the solution set or the truth set of a statement or a predicate is the set of all values that satisfy it. If there is no solution, the solution set is the empty set. [2]
In mathematics, the term undefined refers to a value, function, or other expression that cannot be assigned a meaning within a specific formal system. [1]Attempting to assign or use an undefined value within a particular formal system, may produce contradictory or meaningless results within that system.
Now we shall check when these solutions are singular solutions. If two solutions intersect each other, that is, they both go through the same point (x,y), then there is a failure of uniqueness for a first-order ordinary differential equation. Thus, there will be a failure of uniqueness if a solution of the first form intersects the second solution.