Different types of math equations table
Short summary of common equation types (algebraic, linear, differential etc.).

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Equations by the number of solutions#

Equation typeNumber of solutionsDescriptionExample equationExample solutions
Contradictory equation0No solution, there is no such x, that satisfies the equation (our problem can't be solved in such way).Show sourcesin(x)=10\sin(x) = 10Show source-
Equation with one solution1One solution, there is exactly one x, that satisfies the equation (our problem can be solved in one unique way).Show sourcex+2=5x + 2 = 5Show sourcex=3x = 3
Equation with many solutions2, 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 sourcex2=4x^2 = 4Show sourcex={2,2}x = \left\{-2, 2\right\}
Identity equationInfinite 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 sourcex+x=2xx + x = 2xShow sourcex={10,1,58,1000,}x = \left\{-10, 1, \frac{5}{8}, 1000, \dots \right\}

Equations by the type of unknown#

Equation typeType of unknown
(what is x)
DescriptionExample equationExample solutions
Number equationNumberWe search for number, that we can put instead of x to get truth.Show source2x+2=52x + 2 = 5Show sourcex=32x = \frac{3}{2}
Functional equationFunctionWe search for function, that we can put instead of f to get truth.Show sourceddxf(x)=2x+1\frac{d}{dx} f(x) = 2x + 1Show sourcef(x)=x2+xf(x) = x^2 + x
Diophantine equationInteger numberWe search for integer number, that we can put instead of x to get truth.Show sourcex2=2xx^2 = 2^xShow sourcex=2x = 2

Number equations by the expression form#

Equation typeExpression containing unknown variable(s)DescriptionExample equationExample solutions
Linear equation (first order)Show sourceax+b=0a x + b = 0x is unknown value, a and b are known coefficients (x is raised up to the first power).Show source2x3=02x - 3 = 0Show sourcex=32x = \frac{3}{2}
Quadratic equation (second order)Show sourceax2+bx+c=0a x^2 + b x + c = 0x is unknown value, a, b and c are known coefficients (x is raised up to the second power).Show source2x23x+1=02x^2 - 3x + 1 = 0Show sourcex={1,12}x = \left\{1, \frac{1}{2}\right\}
Polynomial equationShow sourceW(x)=anxn+an1+an2xn2++a1x+a0=0W(x) = a_n x^n + a_{n-1} + a_{n-2} x^{n-2} + \dots + a_1 x + a_0 = 0x is unknown value, an 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 source2x5+x42x1=02x^5 + x^4 - 2x-1 = 0Show sourcex={12,1,i,i,1}x = \left\{-\frac{1}{2}, -1, -i, i, 1\right\}
Logarithmic equationShow sourcea log(x)b=0a\ \log(x) - b = 0Unknown value x is located under logarithm function (equation contains logarithm of unknown value).Show sourcelog(x)=2\log(x) = 2Show sourcex=100x = 100
Exponential equationShow sourceaxb=0a^x - b = 0Unknown value x is located in exponent (equation contains powering to the unknown number).Show source2x=82^x = 8Show sourcex=3x = 3
Trigonometric equationShow source-Unknown value is located under on of trigonometric function such as sine or tangent (equation contains trigonometric function of unknown argument).Show sourcesin(x)=1sin(x) = 1Show sourcex={,4π,2π,2π,4π,}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=52 + x = 5
    is number 3 because when we replace x by it we get:
    2+3=52 + 3 = 5
    5=55 = 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,
  • 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+1x = 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):
      ddxf(x)=2x4\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=0ax + b = 0
    • quadratic equation - x is an unknown, a, b and c are known parameters:
      ax2+bx+c=0ax ^ 2 + bx + c = 0
    • polynomial equation - x is an unknown, polynomial coefficients are known:
      W(x)=anxn+an1xn1+an2xn2a1x+a0=0W(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:
      axb=0a^x - b = 0
    • logarithmic equation - unknown x occurs under the logarithm:
      log(x)b=0log(x) - b = 0
    • etc.
  • Equations can also be divided due to the amount of unknowns:
    • equations with one unknown, e.g.
      x=x+1x = x + 1
    • equation with two unknowns, e.g.
      x+y=3x + 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(x1)+2=2x2(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)=10sin(x) = 10
      (there is no such number whose sine is 10)

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