algebra_equations.theory.txt

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Equation:
  - two equal expression, i.e. X = Y

Unknown:
  - variable in equations
  - often noted with single letter, often x y z ...
  - unknown constant is often noted a b c ...

Indeterminate:
  - often noted X Y Z ...
  - variable that:
     - cannot be known
     - no algebraic property is known
  - handled like a constant in equations

Degree of freedom:
  - often noted v
  - number of variables independent from each other in an equation
  - 0 for constant

Equation solving:
  - analytic: using algebraic rules
  - numeric: approximating iteratively

Analytic equation solving:
  - written as series of equations
  - each step applies same f() to each side
  - goal is to reduce both sides as much possible
     - lower degree
     - fewer variables
     - fewer operations
     - smaller numbers
  - solution:
     - when one side is variable alone
        - usually left-side
        - named dependent variable
     - free|abitrary variables are on other side
        - degrees of freedom:
           - number of free variables
           - particular solution: when 0
           - general|parameterized solution: when > 0
  - function root|zero:
     - also named zero|trivial solution
     - when one side is 0
        - usually right-side
     - can be converted back|from solution, so often used synonymously

System of equations:
  - m equations sharing same variables
  - homogenous: when each equation is a function root
  - redundant|dependent equation:
     - when can be derived, in the the same system, either:
        - equivalent equation: from another equation (i.e. same root)
        - from combining several equations
     - does not add information to the system
     - independent: inverse
  - inconsistent equation:
     - when leading to contradiction
     - cannot be solved
     - consistent: inverse
  - solution set:
     - when each equation is solved
     - noted (VAR,...) = (EXPR,...)
        - meaning VAR = EXPR, VAR2 = EXPR2, ...
  - if n variables, a system of m independent consistent equations is:
     - undeterminated:
        - n > m
        - solution's right-side is not constant
     - exactly determinated:
        - n = m
        - solution's right-side is constant
     - overdeterminated:
        - n < m
        - no solution
  - free variables:
     - only in undeterminated systems
     - degrees of freemdom is n - m
     - free variables are the unsolved ones
        - i.e. can choose
     - in solution set, written as (...,VAR,...) = (...,VAR,...)
  - solving method:
     - substitution:
        - solve each equation independently
        - for each equation, iteratively until cannot reduce anymore:
           - replace variable by its right-side on the other equations
           - reduce other equations
     - merging:
        - for each variable, iteravely until cannot reduce anymore:
           - reduce each equation to prepare for next step
           - apply one equation as operand to other equations
              - usually operation is subtraction
                 - e.g. with polynomials, reduce each equation so they all specify "+ ax^n" with same a|n
     - can be mixed

Graph:
  - representing graphically a solved equation or a solution set
  - each axis|dimension is a solved variable
  - each equation is a n-dimensional plane
  - system solution is intersection of all equations
  - for polynomials: each degree adds a turn (change of direction), for each dimension

Polynomial:
  - if degree n <= 4, has a function root
  - solutions:
     - univariate, to solve x:
        - ax + b = -b/a - x
        - ax^2 + bx + c = ((b^2 - 4ac)^1/2 - b)/2a - x
           - discriminant is b^2 - 4ac
              - if >0, two solutions
              - if 0, one solution
              - if <0, two solutions, complex
  - discriminant:
     - expression that determines the property of roots of a polynomial based on its coefficients