Simultaneous equations

From Academic Kids

In mathematics, simultaneous 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.

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The two example equations intersect twice. Therefore, there are two solutions.

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.

In general, systems of simultaneous equations are extremely 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>

As a general rule of thumb, if there are fewer equations than variables, there will typically be infinitely many solutions, or none. If the number of equations is the same as the number of variables, and the equations are independent, there will typically be finitely many solutions (as in the above example). If there are more independent equations than variables, there will usually be no solutions at all. Therefore systems are frequently considered where the number of variables and independent equations is the same.

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 successiveład_równań


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