Specifically, the flat for the first system can be obtained by translating the linear subspace for the homogeneous system by the vector p. This reasoning only applies if the system Ax = b has at least one solution.
The following computation shows Gauss-Jordan elimination applied to the matrix above: The last matrix is in reduced row echelon form, and represents the system x = −15, y = 8, z = 2. But this also reflects that, in general, no such method can exist: some problems are known to be unsolvable by an algorithm, such as Hilbert's tenth problem, which was proved unsolvable in 1970.
However, depending on the function, the inverse may be difficult to be defined, or may not be a function on all of the set B (only on some subset), and have many values at some point. Here, "in general" means that a different behavior may occur for specific values of the coefficients of the equations. {\displaystyle A\mathbf {x} =\mathbf {b} } Polynomial equations of degree up to four can be solved exactly using algebraic methods, of which the quadratic formula is the simplest example. In general, inconsistencies occur if the left-hand sides of the equations in a system are linearly dependent, and the constant terms do not satisfy the dependence relation. as previously stated, where 6 The number of vectors in a basis for the span is now expressed as the rank of the matrix. x )
The distinction between known variables and unknown variables is generally made in the statement of the problem, by phrases such as "an equation in x and y", or "solve for x and y", which indicate the unknowns, here x and y. Different choices for the free variables may lead to different descriptions of the same solution set. When an equation contains several unknowns, and when one has several equations with more unknowns than equations, the solution set is often infinite. If the solution set is empty, then there are no values of the unknowns that satisfy simultaneously all equations and inequalities. b x x For example, the equation, can be rewritten, using the identity tan x cot x = 1 as, The solutions are thus the solutions of the equation tan x = 1, and are thus the set.
This method generalizes to systems with additional variables (see "elimination of variables" below, or the article on elementary algebra. The solutions of linear equations will generate values, which when substituted for the unknown values, make the equation true.
For example, the equations. However, a linear system is commonly considered as having at least two equations.
−
where A solution to a linear system is an assignment of values to the variables such that all the equations are simultaneously satisfied.
This equation can be viewed as a Diophantine equation, that is, an equation for which only integer solutions are sought. , 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, Now, if we apply the inverse function to both sides of h(x) = c, where c is a constant value in B, we obtain. {\displaystyle A^{-1}} One particular solution is x = 0, y = 0, z = 0. In other words, a solution is a value or a collection of values (one for each unknown) such that, when substituted for the unknowns, the equation becomes an equality. 11 6 An equation may be solved either numerically or symbolically. 3 The following pictures illustrate this trichotomy in the case of two variables: The first system has infinitely many solutions, namely all of the points on the blue line. , …
For solutions in an integral domain like the ring of the integers, or in other algebraic structures, other theories have been developed, see Linear equation over a ring. Using the two fundamental rul… The equations of a linear system are independent if none of the equations can be derived algebraically from the others. , and substituting this back into the equation for 12 If every vector within that span has exactly one expression as a linear combination of the given left-hand vectors, then any solution is unique. There is also a quantum algorithm for linear systems of equations.[7]. There is a close relationship between the solutions to a linear system and the solutions to the corresponding homogeneous system: Specifically, if p is any specific solution to the linear system Ax = b, then the entire solution set can be described as. 0 1 A If there is only one solution, one says that it is a double root.