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In mathematics, simultaneous equations , or systems of equations, are a set of equations containing multiple variables. To solve simultaneous equations, the solver needs to use the provided equations to find the exact value of each variable. The solver may then use a graphical method (by plotting both lines in the same graph and finding the exact coordinates of their intersection), by the matrix method, or by substitution and/or elimination.

Consider the following set of equations:

<math>

\begin{cases} x^2 + y^2 = 1\\ 2x + 4y = 0 \end{cases} </math>

When plotted in a graph, the first equation will appear as a circle, while the second will be a straight line. For equations of this nature (linear equation with a curve), there are three possible solution types:

• The line does not intersect the circle at all. This way, there are no solutions
• The line is a tangent to the circle. This way, there is only one pair of solutions (the only intersection).
• The line cuts the circle. There are two points of intersection, and thus, there are two pairs of solutions.

This will be further demonstrated in this article.

Contents 1 Finding solutions 1.1 Substitution method 1.2 Elimination method 1.3 Matrices

[edit] Finding solutions

If there are fewer independent equations than variables, not all variables can be solved for, and so an answer for one variable must be expressed in terms of other variables. In general, to solve a system, one equation is needed for each unknown variable that needs to be solved. If the number of independent equations is the same as the number of variables, then the system should be solvable. Therefore systems are frequently considered where the number of variables and independent equations is the same.

Because of the importance of this, the phrase in the form "x equations, y unknowns" (for example "2 equations 3 unknowns" or "4 equations, 4 unknowns") is often used to describe systems of equations. If y=x or y<x, then every variable will have an explicit solution set - usually finite.

Systems can have three types of solutions: 1. Systems that represent intersecting set of points (lines, curves, etc.) will have a finite number of specific solutions, each representing a coordinate of intersection. 2. Systems that simplify down to nonsense expressions such as 1 = 0. Such systems contain no real points of overlap. 3. Systems that simplify down to self-evident expressions such as 6=6 or 0=0. Such systems represent the complete overlap of the same lines (curves, points, etc.), and the equations are therefore equivalent. (One can typically be transformed into the other through algebraic manipulation.)

[edit] Substitution method

Systems of simultaneous equations can be hard to solve unless a systematic approach is used. A common technique is the substitution method: Isolate the variable in one of the equations, and substitute that expression where that variable appears in another equation.

In the above example, we first solve the second equation for x:

<math>x = -2y\,</math>

and substitute this result into the first equation:

<math>(-2y)^2 + y^2 = 1\,</math>

After simplification, this yields

<math>y = \pm \sqrt{1 \over 5}</math>

and from x = −2y we obtain the corresponding x values. Our system of equations has two sets of solutions:

<math>x = -2\sqrt{1 \over 5},\ y=\sqrt{1 \over 5} \qquad\mbox{and}\qquad x = 2\sqrt{1 \over 5},\ y=-\sqrt{1 \over 5}\,</math>

[edit] Elimination method

The other commonly used method is generally used to solve simultaneous linear equations. It uses the general principles that each side of an equation still equals the other when both sides are multiplied (or divided) by the same quantity, or when the same quantity is added (or subtracted) from both sides. In multiplication/division, a factor is chosen so that when both sides have equivalent quantities added from another equation in the system (that is, the equations are added), one or more of the variables disappear, the resulting equations are still valid representations in the system, and their smaller number of remaining unknowns thus makes the system of equations easier to solve. As the equations grow simpler through the elimination of some variables, a variable will eventually appear in fully solvable form, and this value can then be "back-substituted" into previously derived equations by plugging this value in for the variable. Typically, each "back-substitution" can then allow another variable in the system to be solved.

[edit] Matrices

Systems of equations may also be represented in terms of matrices, allowing various principles of matrix operations to be handily applied to the problem. Systems of simultaneous linear equations are studied in linear algebra and can always be solved; one uses Gaussian elimination or the Cholesky decomposition. To solve general systems numerically on a computer, the n-dimensional Newton's method may be used. Algebraic geometry is essentially the theory of simultaneous polynomial equations. The question of effective computation with such equations belongs to elimination theory.

Simultaneous equation models are a form of statistical model in the form of a set of linear simultaneous equations. They are often used in econometrics.

In modular arithmetic, simple systems of simultaneous congruences can be solved by the method of successive substitution.da:Substitutionsmetoden it:sistema di equazioni pl:Układ równań sv:Ekvationssystem