Two other solutions are x = 3, y = 6, z = 1, and x = 8, y = 9, z = 2.In fact, this particular set of solutions describes a plane in three-dimensional space, which passes through the three points with these coordinates.For solving larger systems, algorithms are used that are based on linear algebra.
A solution is an assignment of expressions to the unknown variables that makes the equality in the equation true.
In other words, a solution is an expression or a collection of expressions (one for each unknown) such that, when substituted for the unknowns, the equation becomes an identity.
However note that in attempting to find solutions for this equation, if we modify the function's definition – more specifically, the function's domain, we can find solutions to this equation.
So, if we were instead to define that the domain of ƒ consists of the real numbers, the equation above has two solutions, and its solution set is (with a, b, c, d, and k real-valued constants) is a hyperplane.
As with all kinds of problem solving, trial and error may sometimes yield a solution, in particular where the form of the equation, or its similarity to another equation with a known solution, may lead to an "inspired guess" at the solution.
If a guess, when tested, fails to be a solution, consideration of the way in which it fails may lead to a modified guess.A solution of an equation is often also called a root of the equation, particularly but not only for algebraic or numerical equations.A problem of solving an equation may be numeric or symbolic.In mathematics, to solve an equation is to find its solutions, which are the values (numbers, functions, sets, etc.) that fulfill the condition stated by the equation, consisting generally of two expressions related by an equality sign.When seeking a solution, one or more free variables are designated as unknowns.Polynomial equations with a degree of five or higher require in general numerical methods (see below) or special functions such as Bring radicals, although some specific cases may be solvable algebraically, for example , which simplifies this to a quadratic equation in z).In Diophantine equations the solutions are required to be integers.If the solution set of an equation is restricted to a finite set (as is the case for equations in modular arithmetic, for example), or can be limited to a finite number of possibilities (as is the case with some Diophantine equations), the solution set can be found by brute force, that is, by testing each of the possible values (candidate solutions).It may be the case, though, that the number of possibilities to be considered, although finite, is so huge that an exhaustive search is not practically feasible; this is, in fact, a requirement for strong encryption methods.The solution set of a given set of equations or inequalities is the set of all its solutions, a solution being a tuple of values, one for each unknown, that satisfies all equations or inequalities.If the solution set is empty, then there are no values x such that the equations or inequalities becomes true simultaneously.