Equation

A mathematical sentence formed by

setting two expressions equal to

each other

Example 1: 3 – 6 = 18

Example 2: 7 + x = 12

Variable

A symbol, usually a letter, that is

used to represent one or more

numbers

Example: In the expression m + 5,

the letter m is the variable

Solution of an Equation

A number that produces a true

statement when substituted for the

variable in an equation.

The number 3 is a solution of the

equation 8 - 2x = 2, because 8 - 2(3) = 2

Properties of Equality

Addition Property of

Equality

a = b

a + c = b + c

Subtraction Property of

Equality

a = b

a - c = b - c

Multiplication Property

of Equality

a = b

a ∙ c = b ∙ c

Division Property of

Equality

a = b 𝒂

𝒄=

𝒃

𝒄

𝒄 ≠ 𝟎

Ratio A comparison of two numbers by

division.

Example: if there is 1 boy and 3 girls you could write the ratio as: 1:3 (for every one boy there are 3 girls)

1:3 can also be represented or written as

1 to 3 OR 1/3

Proportion An equation that states two ratios (or

fractions) are equal.

Scale Drawing

A drawing that uses a scale to

represent an object as smaller or

larger than the original object.

Scale The ratio of the length in a drawing (or model)

to the length of the real thing.

Example: in the drawing anything with the size of "1"

would have a size of "10" in the real world, so a

measurement of 150mm on the drawing would be

1500mm on the real horse.

Scale Model

A three-dimensional model that uses

a scale to represent an object as

smaller or larger than the actual

object.

Dimensional Analysis

A process that uses rates to

convert measurements from one

unit to another.

Example: 12 pints is equivalent to

how many quarts?

𝟏𝟐 𝒑𝒕 (𝟏 𝒒𝒕

𝟐 𝒑𝒕) =

𝟏𝟐

𝟐𝒒𝒕 = 𝟔 𝒒𝒕

Rate

A ratio that compares two

quantities measured in different

units.

Example: 𝟓𝟓 𝒎𝒊𝒍𝒆𝒔

𝟏 𝒉𝒐𝒖𝒓= 𝟓𝟓 𝒎𝒊/𝒉

Conversion Factor

The ratio of two equal quantities,

each measured in different units.

Example: 𝟏𝟐 𝒊𝒏𝒄𝒉𝒆𝒔

𝟏 𝒇𝒐𝒐𝒕

Precision

The level of detail of a

measurement, determined by the

unit of measure.

Example: A ruler marked in

millimeters has a greater level of

precision than a ruler marked in

centimeters.

Accuracy

The closeness of a given

measurement or value to the actual

measurement or value.

Example: You can find the accuracy of a

measurement by finding the absolute

value of the difference between the

actual and measured values.

Significant Digits

The digits used to express the

precision of a measurement.

Examples:

0.0481 has 3 significant digits

12,000 has 2 significant digits

150.000 has 6 significant digits

Expression

A mathematical phrase that

contains operations, numbers,

and/or variables.

Example: 6x + 1

Term of an Expression In Algebra a term is either:

* a single number, or

* a variable, or

* numbers and variables multiplied

together.

Coefficient

In any term, the coefficient is the

numeric factor of the term or the

number that is multiplied by the

variable.

Example: 3x 3 is the coefficient of x

Constant

A fixed value. In algebra, a constant is a

number on its own, or sometimes a

letter such as a, b or c to stand for a

fixed number.

Numerical Expression

A mathematical phrase that

contains only numbers and

operations.

Example: 9 - 3

Algebraic Expression

An expression that includes at least

one variable. Also called a variable expression.

Examples: 5n, 6 + c, and

8 - x

Equivalent Expressions

Two algebraic expressions are said

to be equivalent if their values

obtained by substituting the values

of the variables are same.

Example: 3(x + 3) = 3x + 9

Literal Equations

An equation that contains two or

more variables.

Examples: 𝒅 = 𝒓𝒕

𝑨 = 𝟏

𝟐𝒉(𝒃𝟏 + 𝒃𝟐)

Inequality An inequality says that two values

are not equal.

Symbol Meaning

< is less than

> is greater than

≤ is less than or equal to

≥ is greater than or equal to

≠ is not equal to

Solution of an Inequality

A number that produces a true

statement when substituted for the

variable in an inequality.

The number 5 is a solution of the

inequality 8 - 2x < 2 because 8 - 2(5) < 2 8 – 10 < 2

-2 < 2 True

Continuous Graph

A graph made up of connected lines

or curves.

Discrete Graph

A graph made up of unconnected

points.

Domain

The set of all inputs of a function.

Range

The set of all outputs of a function.

Set Notation

Notation that includes braces to

describe the elements in a set.

Example: Another way of saying x < 3,

is to use the set notation:

{x x is a real number and x < 3} or

{x x, x < 3}

Function A rule or correspondence which

associates each number x in a (set A) to

a unique number f(x) in a (set B).

Vertical Line Test If a vertical line intersects the

relation's graph in more than one

place, then the relation is a NOT a

function.

Independent Variable

The input variable of a function.

Example:

In the function equation y = x + 3,

x is the independent variable.

Dependent Variable

The output variable of a function.

Example:

In the function equation y = x + 3,

y is the independent variable.

Function Notation

A way to name a function using the

symbol f(x) instead of y. The symbol

f(x) is read as “the value of f at x” or

as “f of x”.

Example: The function y = 2x – 9

can be written in function notation

as f(x) = 2x – 9.

Combine Like Terms

Combine all constants into one term

and all terms with the same variable

into one term.

Example:

3x + 2 - 2x + 9

is simplified as x + 9

Distributive Property

A property can be used to find the

product of a number and a sum or

difference.

Example:

3(x + 4) = 3(x) + 3(4)

𝑎(𝑏 + 𝑐) = 𝑎𝑏 + 𝑎𝑐 (𝑏 + 𝑐)𝑎 = 𝑏𝑎 + 𝑐𝑎 𝑎(𝑏 − 𝑐) = 𝑎𝑏 − 𝑎𝑐 (𝑏 − 𝑐)𝑎 = 𝑏𝑎 − 𝑐𝑎

Sequence

A list of numbers in a specific order

that often form a pattern.

Example:

Term of a Sequence

An element or number of a

sequence.

Example:

Explicit Rule

A formula that defines the nth term

an, or general term, of a sequence as

a function of n. Explicit rules can be

used to find any specific term in a

sequence without finding the

previous terms.

Example: f(n) = 2n – 9.

Recursive Rule

A formula for a sequence in which

one or more previous terms are used

to generate the next term.

Example: f(1) = 4,

f(n) = f(n – 1) + 10

The sequence for this recursive rule is

created using the sum of the previous

term f(n – 1) and 10.

Arithmetic Sequence A sequence whose successive terms

differ by the same nonzero number d,

called the common difference. It can be

described by an explicit or recursive

rule.

Example: f(n) = 2000 + 500(n – 1) or f(1)

= 2000, f(n) = f(n – 1) + 500 for n ≥ 2;

both have a common difference of 500

and the first term is 2000.

Common Difference

In an arithmetic sequence, the

nonzero constant difference of any

term and the previous term.

Example: The arithmetic sequence:

5, 9, 13, 17, … has a common

difference of 4.

x-intercept (Zero)

Location where

a straight line

crosses the x-

axis of a graph;

the location is

represented by

an ordered pair

(x, y).

y-intercept

Location where

a straight line

crosses the y-

axis of a graph;

the location is

represented by

an ordered pair

(x, y).

Rate of Change A comparison of a change in one

quantity with a change in another

quantity. In real-world situations,

you can interpret the slope of a line

as a rate of change.

Example: 𝒄𝒉𝒂𝒏𝒈𝒆 𝒊𝒏 𝒄𝒐𝒔𝒕

𝒄𝒉𝒂𝒏𝒈𝒆 𝒊𝒏 𝒕𝒊𝒎𝒆

Slope The slope of a non-vertical line is the

ratio of the vertical change (the rise) to

the horizontal change (the run) between

any two points (x1, y1) and (x2, y2).

Slope is indicated by the letter m

𝑚 =

𝑟𝑖𝑠𝑒

𝑟𝑢𝑛 =

𝑦2−𝑦1

𝑥2−𝑥1 =

∆𝑦

∆𝑥 =

𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑦

𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑥

Slope

Example: the slope is 3/5

Number of units

up or down

Number of units

left or right

The change in y means you will

move up 3 units since 3 is positive; The change in x

means you will move right 5 units since 5 is positive.

x

y

=

Slope Facts 1. Horizontal lines have a slope of

zero; m=0

2. Vertical lines have undefined slope

3. Parallel lines have the same slope

4. Perpendicular lines have slopes that

are opposite reciprocals. Their

product is -1. If m1 and m2 are the

slopes of two perpendicular lines, then

m1m2 =-1

Direct Variation Two variables x and y show direct

variation provided that y = ax, where a

is a nonzero constant. The variable y

is directly proportional to x.

Example: Speed and Distance d = 60t

The distance traveled is directly

proportional to the amount of time

traveled.

Constant of Variation

The number that relates two

variables that are directly

proportional. The nonzero

constant a in a direct variation

equation y = ax.

Example: y = -2x

constant of variation

Slope-Intercept Form

Used when you have the slope and

the y-intercept.

y = mx + b slope y-int.

Linear Function

The equation Ax + By = C represents

a linear function provided B ≠ 0.

Example: The equation 2x – y = 3

represents a linear function. The

equation x = 3 does not represent a

function.

Linear Equation An equation that makes a straight

line when it is graphed.

y = mx + b

m = slope

b = y-intercept

Standard Form of a

Linear Equation

A linear equation written in the form

Ax + By = C, where A and B are not both

zero.

Example: The linear equation y = -3x +

4 can be written in standard form as

3x + y = 4.

Solution of a Linear

Equation in Two Variables An ordered pair that produces a true

statement when the coordinates of

the ordered pair are substituted for

the variables in the equation.

Example: (1, -4) is a solution of 3x –

y =7, because 3(1) – (-4) = 7.

Discrete Function A function with a graph that

consists of isolated points.

Continuous Function A function with a graph that is

unbroken.