🏵️ Pʀᴇ-Aʟɢᴇʙʀᴀ

🚦 Variable

🚥 A variable is a symbol, sigil, letter, etc... that represents and stands for something else.

🚥 Usually a number, it can stand for a point, a function, a line, a relationship, a shape, a quantity, a direction, a probability, an interval, an object, a subject, a string... so many things can be variables!

🧭 Algebraic Properties

🎚️ Monomial

🎚️ A single term that represents a concept. x, 3a, 4

🕰️ Binomial

🕰️ Two terms added into an expression 2+a, 3a-b, x²+y²

Equalities

An Equality is a relationship between two concepts expressed by an expression. Both sides of the equality have an equivalent value.

\color{#CC444B}  
A = B

\color{#DA5552}  
⟹ A + C = B + C

\color{#DF7373}  
⟹ A · C = B · C

Linear Equation

Lineal equations describe a straight line. We solve the value of that line at the origin.

\color{#DF7373}  
ax + b = 0

Cuadratic Equation

Lineal equations describe a parabola. We solve the values at the origin.

\color{#FF6666}  
ax² + bx + c = 0

\color{#FF6055}  
x= \frac{-b±\sqrt{b²-4ac}}{2a}

Inequalities

A ≤ B

⟹ A + C ≤ B + C

⟹ c·A  ≤ c·B  ⠀⠀c∈ℝ⁺

⟹ c·A  ≥ c·B  ⠀⠀c∈ℝ⁻

⟹ A>0　B>0 　⟶ 　A≤B ⟺ \frac{1}{A} ≥ \frac{1}{B}  ⠀

 A≤B⠀C≤D ⠀⟹ 　A+C ≤⠀B+D

Algebraic roots

🎞️ Polynomials

🎚️ A Polynomial on the variable $𝑥$ is an expression of the form:

𝑎₀ + 𝑎₁ 𝑥 + 𝑎₂ 𝑥² + 𝑎₃ 𝑥³ .+. 𝑎ₙ 𝑥ⁿ ... 

🎚️ That we can express

∑ 𝑎ₙ 𝑥ⁿ

🎛️ The Polynomial has a grade n when its exponent goes up to $n$.

🎛️ Every monomial summed is called a term.

Solving Systems Of Equations

In a system of linear equations of the form:

\left\{\begin{matrix}
a_{11}x_1 + a_{12}x_2 +a_{13}x_3 + ... + a_{1n}x_n  = b_1 \\
a_{21}x_1 + a_{22}x_2 +a_{23}x_3 + ... + a_{2n}x_n  = b_2 
\\
 ... \\
a_{m1}x_1 + a_{m2}x_2 +a_{m3}x_3 + ... + a_{mn}x_n  = b_m 
\end{matrix}\right.

where $a_{ij}$ are the coefficients, $b_i$ the known factors, and $x_j the unknowns.

If $∀b = 0$ the system is homogeneous.

A _solution is a n-tuple $（s_1,s_2,...,s_n）$ that satisfy all equations.

• A system is incompatible if it admits no solutions for all equations simultaneously.

\begin{Bmatrix}
x_1 + x_2 = 3 \\
x_1 + x_2 = 1
\end{Bmatrix}

• A system is said to be compatible if it admits solutions.
  • Two equations are independent if no equation is linear combination of another.
  • If a system of n equations and n unknowns is independent it has only one touple of solutions.
  • If a system has dependent equations or fewer equations than unknowns then it can have an infinite set of solutions.

\begin{Bmatrix}
x_1 + x_2 = 3 \\
2x_1 + 2x_2 = 6
\end{Bmatrix}

Two linear systems are called equivalent if they have the same solutions.