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.)
dummies: Algebra I Cheatsheat
dummies: Algebra II Cheatsheat
\begin{align} a = b \end{align}
We can transform an equation many times and many ways, so long as we maintain the equality.
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}
Simplify expressions
\begin{align} & x+x+x\\ & 3x \end{align}
\begin{align} & x*x*x\\ & x^{3} \end{align}
\begin{align} & 2x^{2}+3ab-x^{2}+ab\\ & x^{2}+4ab \end{align}
\begin{align} & x(2x+3)\\ & (x*2x)+(x*3)\\ & 2x^{2}+3x \end{align}
\begin{align} & 6x^{5}+3x^{2}\\ & 3x^{2}(2x^{3}+1) \end{align}
\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}
in functional notation: $a = b \Rightarrow f(a) = f(b)$
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.
ax + b = y
Solving equations.
Express the problem as a set of equations.
Multiple ways to solve.
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.