## Key Concepts

- The addition property of equality states, for all real numbers [latex]a, b, \text{ and } c: \text{ if } a=b \text{, then } a+c=b+c[/latex]. 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 [latex]a, b, \text{ and } c: \text{ if } a=b \text{, then } a \cdot c=b \cdot c[/latex]. 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 [latex]\left(x, y\right)[/latex] where [latex]x[/latex] is called the x-coordinate and [latex]y[/latex] 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 [latex]m=\dfrac{\text{rise}}{\text{run}}[/latex], 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

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

## 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 [latex]1[/latex]
**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*.