Summary: Review Topics

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.

[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].

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.