Short summary of common equation types (algebraic, linear, differential etc.).

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Equations by the number of solutions#

Equation type	Number of solutions	Description	Example equation	Example solutions
Contradictory equation	0	No solution, there is no such x, that satisfies the equation (our problem can't be solved in such way).	Show source $\sin(x) = 10$	Show source $-$
Equation with one solution	1	One solution, there is exactly one x, that satisfies the equation (our problem can be solved in one unique way).	Show source $x + 2 = 5$	Show source $x = 3$
Equation with many solutions	2, 3, 4...	Many number of solutions, there are more than one x (2,3,4 etc.), that satisfy the equation (our problem can be solved in more than one way).	Show source $x^2 = 4$	Show source $x = \left\{-2, 2\right\}$
Identity equation	∞	Infinite number of solutions, no matter what x is, the equation is always satisfied (congratulations ☺, you've just discover another rule, that rules our universe).	Show source $x + x = 2x$	Show source $x = \left\{-10, 1, \frac{5}{8}, 1000, \dots \right\}$

Equations by the type of unknown#

Equation type	Type of unknown (what is x)	Description	Example equation	Example solutions
Number equation	Number	We search for number, that we can put instead of x to get truth.	Show source $2x + 2 = 5$	Show source $x = \frac{3}{2}$
Functional equation	Function	We search for function, that we can put instead of f to get truth.	Show source $\frac{d}{dx} f(x) = 2x + 1$	Show source $f(x) = x^2 + x$
Diophantine equation	Integer number	We search for integer number, that we can put instead of x to get truth.	Show source $x^2 = 2^x$	Show source $x = 2$

Number equations by the expression form#

Equation type	Expression containing unknown variable(s)	Description	Example equation	Example solutions
Linear equation (first order)	Show source $a x + b = 0$	x is unknown value, a and b are known coefficients (x is raised up to the first power).	Show source $2x - 3 = 0$	Show source $x = \frac{3}{2}$
Quadratic equation (second order)	Show source $a x^2 + b x + c = 0$	x is unknown value, a, b and c are known coefficients (x is raised up to the second power).	Show source $2x^2 - 3x + 1 = 0$	Show source $x = \left\{1, \frac{1}{2}\right\}$
Polynomial equation	Show source $W(x) = a_n x^n + a_{n-1} + a_{n-2} x^{n-2} + \dots + a_1 x + a_0 = 0$	x is unknown value, a_n are known polynomial coefficients (it's a generalization of linear and quadratic equations to polynomial of any range, x can be raised to any power).	Show source $2x^5 + x^4 - 2x-1 = 0$	Show source $x = \left\{-\frac{1}{2}, -1, -i, i, 1\right\}$
Logarithmic equation	Show source $a\ \log(x) - b = 0$	Unknown value x is located under logarithm function (equation contains logarithm of unknown value).	Show source $\log(x) = 2$	Show source $x = 100$
Exponential equation	Show source $a^x - b = 0$	Unknown value x is located in exponent (equation contains powering to the unknown number).	Show source $2^x = 8$	Show source $x = 3$
Trigonometric equation	Show source $-$	Unknown value is located under on of trigonometric function such as sine or tangent (equation contains trigonometric function of unknown argument).	Show source $sin(x) = 1$	Show source $x = \left\{\dots, -4\pi, -2\pi, 2\pi, 4\pi, \dots\right\}$

Some facts#

In the equation some part is known, and another one is unknown.
Solving the equation consists of finding what to substitute in place of the unknown to get the true sentence (equality) . We say then that solution satisfies the equation.

ⓘ Example: The solution of the equation:
$2 + x = 5$
is number 3 because when we replace x by it we get:
$2 + 3 = 5$
so:
$5 = 5$
Equations generally reflect some practical problems in which we seek a value that meets our criteria e.g.:
- we know that 6 eggs are needed for one cheesecake, we want to know how many eggs we need to bake two cheesecakes for the upcoming holidays,
- we need to get to a city 300 km away, we wonder how long time it will take us on a trip if we were going on average at 50 km/h,
- we have pocket money of 10$ per week and we would like to buy a bike for 500$, we wonder how many weeks we have to put off to buy it,
  [Li]etc.
Equations can be divided due to the type of object we are looking for (type of solution):
- number equations - if the solution is the number (e.g. number 3), we are looking for answers to questions like:
  - "how much"
  - "how many",
  - "at what price",
  - "is there a number that...",
  - etc.
  Example number equation is (x is an unknown number):
  
  $x = x + 1$
- functional equations - if solution is a function (e.g. f(x) = 3x² - 5), we are looking for answers to questions such as:
  - "is there a function that ...",
  - "after which the track will move ...",
  - "how one value depends on another one",
  - "the derivative of what function is ...",
  - etc.
  An example of a functional equation is (f is an unknown function):
  
  $\dfrac{d}{dx} f(x) = 2x-4$
- etc.
Number equations are often divided due to type of expression that they contain, for example:
- linear equation - x is an unknown, a and b are known parameters:
  
  $ax + b = 0$
- quadratic equation - x is an unknown, a, b and c are known parameters:
  
  $ax ^ 2 + bx + c = 0$
- polynomial equation - x is an unknown, polynomial coefficients are known:
  
  $W(x) = a_n x^n + a_{n-1} x^{n-1} + a_{n-2} x^{n-2} \dots a_1 x + a_0 = 0$
- exponential equation - unknown x occurs in the exponent of power:
  
  $a^x - b = 0$
- logarithmic equation - unknown x occurs under the logarithm:
  
  $log(x) - b = 0$
- etc.
Equations can also be divided due to the amount of unknowns:
- equations with one unknown, e.g.
  
  $x = x + 1$
- equation with two unknowns, e.g.
  
  $x + y = 3$
- etc.
Equations can have more than one solution (our problem can be solved in different ways) or has no solutions (our problem can't be solved in that way). Depending on the number of solutions, the equation can be divided into:
- identity equation - there are infinitely many of solutions e.g.
  
  $2(x - 1) + 2 = 2x$
  (we can find any number of different numbers that when can substituted for x and get the true sentence)
- contradictory equations - there is no solution, e.g.:
  
  $sin(x) = 10$
  (there is no such number whose sine is 10)

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