Simultaneous equations

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In mathematics, simultaneous equations, or systems of equations, are a set of equations where variables are shared. A solution consists of values for the variables which satisfy all of the equations simultaneously.

For example, the following is a set of equations:

<math>x^2 + y^2 = 1\,</math>

<math>2x + 4y = 0\,</math>

which geometrically describes the intersection of a circle and a straight line.

Finding solutions

To get a numerical solution to an unknown, one must have at least as many independant equations as variables. If there are fewer independant equations than variables, there will be infinitely many solutions - another way to say this is that the solution set will be an infinite set (the set containing these infinitely many solutions). When this happens, in many cases, it is said that there are no solutions to the particular unknown one is looking for - however this is a misnomer. In such a case the solutions to an unknown depend on other variables.

If the number of independant equations is the same as the number of variables, there will typically be finitely many solutions. Therefore systems are frequently considered where the number of variables and independent equations is the same.

Because of the importants 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.

Substitution method

Image:Simultaneous equations example 1.png

Systems of simultaneous equations can be hard to solve. A common technique is the substitution method: try to solve one of the equations for one of the variables and substitute the result into the other equations, thereby reducing the number of equations and the number of variables by 1. Continue until you reach a single equation with a single variable, which (hopefully) can be solved; back substitution then yields the values for the other variables.

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 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>

Systems of simultaneous linear equations are studied in linear algebra and can always be solved; one uses Gauss-Jordan 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.pl:Układ równań sv:Ekvationssystem