======Algebra======

In [[arithmetic]] we learned the
arithmetic operations: addition, subtraction, multiplication, division, 
and exponentiation.

In **algebra** we learn to apply those operations in **equations**.

The word algebra can be traced back to the Persian al-jabr.  
In a book written by ninth century mathematician and astronomer
al-Khawarizmi, the term al-jabr refers to the operation of 
moving a term from one side of an equation to the other.  
(Note: this will be illustrated below.)

[[https://www.dummies.com/education/math/algebra/algebra-i-for-dummies-cheat-sheet/
|dummies: Algebra I Cheatsheat]] \\
[[https://www.dummies.com/education/math/algebra/algebra-ii-for-dummies-cheat-sheet/
|dummies: Algebra II Cheatsheat]]
====Equation====

\begin{align}
a = b
\end{align}

We can transform an equation many times and many ways, so long as we maintain the equality.


====Properties====

Commutative

\begin{align}
a+b &= b+a && \text{addition} \\
ab &= ba && \text{multiplication} \\
\end{align}

Associative

\begin{align}
(a+b)+c &= a+(b+c) && \text{addition} \\
(ab)c &= a(bc) && \text{multiplication} \\
\end{align}

Distributive

\begin{align}
& a(x+y)\\
& ax + ay \\
\end{align}



====Variables and Expressions====

Simplify expressions
  * Put added terms together using coefficients.
\begin{align}
 & x+x+x\\
 & 3x
\end{align}

  * Put multiplied terms together using exponents.
\begin{align}
 & x*x*x\\
 & x^{3}
\end{align}

  * Add like terms together.
\begin{align}
 & 2x^{2}+3ab-x^{2}+ab\\
 & x^{2}+4ab
\end{align}

  * Multiply bracketed terms using the distributive property. 
\begin{align}
 & x(2x+3)\\
 & (x*2x)+(x*3)\\
 & 2x^{2}+3x
\end{align}

  * Factor.  Reverse-multiply bracketed terms. 
\begin{align}
 & 6x^{5}+3x^{2}\\
 & 3x^{2}(2x^{3}+1)
\end{align}

====Equations====

  * Apply an operation on both sides of an equation, and the equality remains.

\begin{align}
a &= b && \text{given }\\
a+c &= b+c && \text{add a constant to both sides }\\
ac &= bc && \text{multiply a constant on both sides } \\
\end{align}


===Functions===

in functional notation: $a = b \Rightarrow f(a) = f(b)$



===Substitution===




===Solving===

Subtract a term from both sides of the equation.

Apply a function to both sides of the equation, systematically, 
to simplify or to isolate a target variable.



=====Linear Equations=====
ax + b = y


Solving equations.

=====System of Equations=====

Express the problem as a set of equations.

Multiple ways to solve.
  * graph
  * elimination
  * substitution

Subtract one equation from another.  
Is the same as subtracting one value from both sides of an equation
because the the two sides are equal.