Equation solver with steps
In math, an equation is an equality that contains one or more variables. The equation is to determine the values that can take the variable to true equality. The variable is also called unknown and the values for which equality is checked solutions. Unlike an identity, an equation is an equality that is not necessarily true for all possible values that can take the variable2, 3.
The Equation solver with steps can be of various types, are found in different branches of mathematics; the techniques associated with their treatment differ according to their type.
Algebra studies mainly two families of equations: equations polynomial and among them linear equations solver. Polynomial equations are of the form P (X) = 0, where P is a polynomial. Methods of transformation and change of variable to cope with the simplest. Linear equations are of the form a (x) + b = 0, where a is a linear and b a vector. Used to solve algorithmic or geometric, from techniques of linear algebra and analysis. Change the domain of the variable can change substantially the nature of the equation. Algebra is also studying Diophantine equations, equations whose coefficients and the solutions are integers. The techniques used are different and mainly from arithmetic. These equations are generally difficult, often only one seeks to determine the existence or the absence of solution and, if they exist, their number.
Geometry uses equations to characterize figures. The goal is still different from the previous cases, the equation is used to highlight the geometric properties. There are, in this context, two main families of equations, the Cartesian and parametric.
The analysis studied equations of the type f (x) = 0, where f is a function with certain properties like continuity, the differentiability or being Contracting. Techniques to build sequences converging to a solution of the equation. The goal is to be able to approach the solution as precisely as possible.
A dynamic system is defined by an equation whose solutions are either suites or one or several variables functions. There are two central issues: the initial state and the asymptotic behavior. For each eligible initial state, for example the value of the suite or function in zero, the equation admits a unique solution. Sometimes, a small change in the initial state changes little the solution. This isn’t always the case, this sensitivity to the initial condition is the subject of the first question.
The behavior limited or even asymptotic of solution corresponds to the form of the solution when the variable approaches infinity, this behavior is the subject of the second question. If it deviates not, he can, either towards a given value, or approaching a cyclical behavior (a periodic function or a suite always browsing the same set of values and in the same order), or have a chaotic behavior, seeming evolve at random, even if the solution is inherently deterministic.
In Euclidean geometry, it is possible to associate to each point in space a set of coordinates, for example using an orthonormal.
This method allows to characterize geometric figures using equations. A plan in a 3 dimensional space is expressed as the set of solutions of an equation of the type ax + by + cz + d = 0, where a, b, c, and d are real numbers, x, y, z the unknowns that correspond to the coordinates of a point of the plan in the orthonormal coordinate system. The values a, b and c are the coordinates of a vector perpendicular to the plane defined by the equation. A right is expressed as the intersection of two planes, as solutions to a linear equation values in ℝ2 or as solutions of a system of two linear equations with values in ℝ, if ℝ designates the set of real numbers.
The Cartesian equation provides a simple method of proof of the theorem of Thales on the circle.
A conic section is the intersection of a cone of equation x 2 + y2 = z2 and plan. In other words, in the space, all Conic is defined as points whose coordinates are solutions of an equation of the plan in ℝ2 and the previous equation. This formalism allows to determine the positions and the properties of the conical homes.
With this approach, we obtain equations whose objective is not the expression of solutions in the sense of the previous paragraph. An example is given by the theorem of Thales indicating that a triangle is right-angled if it has one side equal to a diameter of a circle and a part of the circle opposite Summit. This theorem is illustrated in the figure at right. If the mark is well chosen, it is orthogonal and the equation of the circle is written: x 2 + y2 = 1, points A and C of the figure on the right have respective coordinates (-1.0) and (1,0). Say that AB is perpendicular to CB is to say that the associated vectors are orthogonal.