equation - math with ms. deleon
TRANSCRIPT
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Equation
A mathematical sentence formed by
setting two expressions equal to
each other
Example 1: 3 – 6 = 18
Example 2: 7 + x = 12
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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
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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
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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
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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
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Proportion An equation that states two ratios (or
fractions) are equal.
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Scale Drawing
A drawing that uses a scale to
represent an object as smaller or
larger than the original object.
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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.
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Scale Model
A three-dimensional model that uses
a scale to represent an object as
smaller or larger than the actual
object.
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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
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Rate
A ratio that compares two
quantities measured in different
units.
Example: 𝟓𝟓 𝒎𝒊𝒍𝒆𝒔
𝟏 𝒉𝒐𝒖𝒓= 𝟓𝟓 𝒎𝒊/𝒉
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Conversion Factor
The ratio of two equal quantities,
each measured in different units.
Example: 𝟏𝟐 𝒊𝒏𝒄𝒉𝒆𝒔
𝟏 𝒇𝒐𝒐𝒕
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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.
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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.
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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
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Expression
A mathematical phrase that
contains operations, numbers,
and/or variables.
Example: 6x + 1
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Term of an Expression In Algebra a term is either:
* a single number, or
* a variable, or
* numbers and variables multiplied
together.
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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
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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.
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Numerical Expression
A mathematical phrase that
contains only numbers and
operations.
Example: 9 - 3
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Algebraic Expression
An expression that includes at least
one variable. Also called a variable expression.
Examples: 5n, 6 + c, and
8 - x
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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
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Literal Equations
An equation that contains two or
more variables.
Examples: 𝒅 = 𝒓𝒕
𝑨 = 𝟏
𝟐𝒉(𝒃𝟏 + 𝒃𝟐)
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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
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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
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Continuous Graph
A graph made up of connected lines
or curves.
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Discrete Graph
A graph made up of unconnected
points.
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Domain
The set of all inputs of a function.
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Range
The set of all outputs of a function.
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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}
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Function A rule or correspondence which
associates each number x in a (set A) to
a unique number f(x) in a (set B).
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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.
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Independent Variable
The input variable of a function.
Example:
In the function equation y = x + 3,
x is the independent variable.
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Dependent Variable
The output variable of a function.
Example:
In the function equation y = x + 3,
y is the independent variable.
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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.
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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
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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
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Sequence
A list of numbers in a specific order
that often form a pattern.
Example:
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Term of a Sequence
An element or number of a
sequence.
Example:
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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.
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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.
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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.
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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.
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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).
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y-intercept
Location where
a straight line
crosses the y-
axis of a graph;
the location is
represented by
an ordered pair
(x, y).
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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
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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
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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
=
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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
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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.
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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
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Slope-Intercept Form
Used when you have the slope and
the y-intercept.
y = mx + b slope y-int.
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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.
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Linear Equation An equation that makes a straight
line when it is graphed.
y = mx + b
m = slope
b = y-intercept
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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.
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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.
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Discrete Function A function with a graph that
consists of isolated points.
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Continuous Function A function with a graph that is
unbroken.