Title: equations

number of equations

Unbenannter Text

Equations can occur as single equations, i.e. there is one equation that leads to true statements by inserting values for the variable(s).

example for a linear equation: \(3x - 5 = x -1\) oder \(2x^4-x^2+1 = 0\)

A system of equations contains several equations. Solutions are then sought so that all equations become true statements when the values for the variable(s) are inserted.

example for a linear system of equations:
\(x - 1 = 0\)
\( 2x-3 = -1\)
This system of equations consists of 2 equations with 1 variable, the only solution is \(x = 1\). Substituting gives true statements: \(1-1 \overset{!}{=} 0 \land 2\cdot 1 - 3 \overset{!}{=} -1\)

A system of equations does not have to be quadratic, i.e. there can be more equations than variables or more variables than equations.

numbers of variables

Unbenannter Text

Equations can contain one or more variables.
If there are more variables than equations in a system of equations, there is no longer a clear solution.

linear equation with one variable:
\(ax = b\)

linear equation with \(n\) variables:
\(a_1\cdot x_1 + a_2\cdot x_2 + \ldots + a_n\cdot x_n = b\)

linear system of equations with 2 variables:

\(a_{11}\cdot x_1 + a_{12}\cdot x_2 = b_1 \)
\(a_{21}\cdot x_1 + a_{22}\cdot x_2 = b_2\)

linear system of equations with \(m\) equations and \(n\) variables:

\(a_{11}\cdot x_1 + a_{12}\cdot x_2 + \ldots + a_{1n}\cdot x_n= b_1 \)
\(a_{21}\cdot x_1 + a_{22}\cdot x_2 + \ldots + a_{2n}\cdot x_n= b_2 \)
\(\vdots\)
\(a_{m1}\cdot x_1 + a_{m2}\cdot x_2 + \ldots + a_{mn}\cdot x_n= b_m \)

(simple) categorisation of equations

Unbenannter Text

Algebraic equations are equations in which the variables can take on any real numbers.

Diophantine equations are equations in which the variables can only have integer solutions.
(Application: cryptography, number distribution, ...)

Differential equations are equations that contain not only the variables but also their derivatives.
(Application: mechanics, epidemology, electrical engineering, astronomy, ...)