Key Concepts

• The addition property of equality states, for all real numbers $a, b, \text{ and } c: \text{ if } a=b \text{, then } a+c=b+c$. That is, we may add or subtract the same amount to both entire sides of an equation without changing its value.
• The multiplication property of equality states, for all real numbers $a, b, \text{ and } c: \text{ if } a=b \text{, then } a \cdot c=b \cdot c$. That is, we may multiply or divide the same amount to both entire sides of an equation without changing its value.
• Any point graphed in the coordinate plane is of form $\left(x, y\right)$ where $x$ is called the x-coordinate and $y$ is called the y-coordinate. With these coordinates, any point in the plane may be unambiguously located or identified.
• The coordinates of any ordered pair contained in the graph of an equation satisfies the equation (makes it a true statement when substituted for x and y).
• The slope of a line, of form $m=\dfrac{\text{rise}}{\text{run}}$, is a measure of the steepness of a line.
• Parallel lines have identical slopes; perpendicular lines have opposite, reciprocal slopes.

Key Expressions, Equations, and Inequalities

• $\displaystyle \text{Slope }=\frac{\text{rise}}{\text{run}}$ and $\displaystyle m=\frac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}$ where $m=\text{slope}$ and $\displaystyle ({{x}_{1}},{{y}_{1}})$ and $\displaystyle ({{x}_{2}},{{y}_{2}})$ are two points on the line.
• For any number x and any integers a and b, $\left(x^{a}\right)\left(x^{b}\right) = x^{a+b}$.
• For any non-zero number x and any integers a and b: $\displaystyle \frac{{{x}^{a}}}{{{x}^{b}}}={{x}^{a-b}}$
• For any positive number x and integers a and b: $\left(x^{a}\right)^{b}=x^{a\cdot{b}}$.
• For any nonzero numbers a and b and any integer x, $\left(ab\right)^{x}=a^{x}\cdot{b^{x}}$.
• For any number a, any non-zero number b, and any integer x, $\displaystyle {\left(\frac{a}{b}\right)}^{x}=\frac{a^{x}}{b^{x}}$.
• Any number or variable raised to a power of 1 is the number itself. $n^{1}=n$
• Any non-zero number or variable raised to a power of 0 is equal to 1. $n^{0}=1$
• For any nonzero real number $a$ and natural number $n$, the negative rule of exponents states that ${a}^{-n}=\frac{1}{{a}^{n}}$.
• With a, b, m, and n not equal to zero, and m and n as integers, the following rules apply: $a^{-m}=\frac{1}{a^{m}}$,   $\frac{1}{a^{-m}}=a^{m}$,   $\frac{a^{-n}}{b^{-m}}=\frac{b^m}{a^n}$.

Glossary

absolute value

a number’s distance from zero on the number line, which is always positive

constant

a number whose value always stays the same

coefficient

a number multiplying a variable

equation

two expressions connected by an equal sign

exponent

a number in a superscript position that tells how many times to multiply the base by itself

expression

a number, a variable, or a combination of numbers and variables and operation symbols

reciprocal

two fractions are reciprocals if their product is $1$

term

a single number, variable, or a product or quotient of numbers and/or variables

variable

a symbol that stands for an unknown quantity, often represented with letters, like x, y, or z.