Algebraic Manipulation Reference Cheat Sheet

Key Facts

• The distributive property is $a(b+c)=ab+ac$ and $a(b-c)=ab-ac$.
• Like terms have the same variable part, so $3x+5x=8x$ but $3x+5y$ cannot be combined.
• To remove parentheses with a negative sign, change every sign inside: $-(2x-7)=-2x+7$.
• Factoring reverses distribution, so $6x+9=3(2x+3)$.
• To solve a linear equation, use inverse operations on both sides to isolate the variable, as in $2x+5=13 \Rightarrow 2x=8 \Rightarrow x=4$.
• The zero product property says if $ab=0$, then $a=0$ or $b=0$.
• When multiplying powers with the same base, add exponents: $x^m \cdot x^n=x^{m+n}$.
• To rearrange a formula, treat the target variable like the unknown and undo operations in reverse order, such as $A=lw \Rightarrow w=\frac{A}{l}$.

Vocabulary

Expression

An expression is a mathematical phrase with numbers, variables, and operations but no equals sign.

Equation

An equation is a statement that two expressions are equal, such as $3x+2=11$.

Like Terms

Like terms are terms with exactly the same variable factors and exponents, such as $4x^2$ and $-7x^2$.

Coefficient

A coefficient is the number multiplying a variable, such as $-5$ in $-5x$.

Factor

A factor is a number or expression multiplied by another factor to make a product.

Equivalent Expressions

Equivalent expressions have the same value for every allowed value of the variable.