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
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\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

• 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}

• 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}

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.

• 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.