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Variables
Tutorial 3c
Variables

A variable is any symbol that can be
replaced with a number to solve a math
problem.

An open sentence has at least one variable.
–
–
An algebraic equation is an open sentence.
An open sentence can be proven neither true
nor false until the variable is replaced with a
number.
Variables

Letters are used to represent numbers, and these
letters are referred to as variables.


For example, in the equation 3 + x = 5, x is the letter
that represents the numeral 2.
However, in 3 + x =
, the x is a variable that could
represent many different numerals depending on the
number placed in the blank.
 For example:



3 + x = 7 The variable x represents the numeral 4.
3 + x = 9 The variable x represents the numeral 6.
3 + x = 50 The variable x represents the numeral 47.
Variables
3 + x = 7 The variable x represents the numeral 4.
3 + x = 9 The variable x represents the numeral 6.
3 + x = 50 The variable x represents the numeral 47.
In
each equation above, the variable is x , and the
number 3 is called a constant because 3
represents the same value in each equation.
The answer following the equal sign depends on
the number assigned to the variable (x).
Variables
Algebraic equations have at least one variable. An
algebraic equation can be proven neither true nor false
until the variable is replaced with a number.
Example:
6+2=x
x must be 8 for this open
sentence to be true
2
x
6
Examples
6+2=x
x must be 8 for this equation to
be true
y = 12 - 6
y must be 6
Find the value of the variable that makes each equation true.
1. A = 6 + 2
A=
4
8
2. 23 - 6 = x
x=
8
13 17
3. G = 12 + 7 G =
4. 21 - 7 = d
10
5 19 31
d = 14 17 28
Addition Properties
Commutative Property of Addition :
The order in which the numbers are
added does not change their sum.
Example :
15 + 3
Is the same as 3 + 15
15 + y
Is the same as y + 15
Addition Properties
Associative Property of Addition:
The way that 3 or more numbers are grouped
does not change their sum. Parentheses () may
be used to show which numbers are added first.
Example :
(8 + 7) + 6
Is the same as 8 + (7 + 6)
(8 + x) + 6
Is the same as (8 + 6) + x
Addition Properties
Additive Identity of Addition:
The sum of any number and zero will always
result in the original number.
Example :
8+0=8
51 + 0 = 51
Choose the correct property.
1. 9 + 0 = 9
Associative
Commutative
Identity
2. 7 + 8 = 8 + 7
Associative
Commutative
Identity
3. (2 + 4) + 3 = 3 + (2 + 4)
Associative
Commutative
Identity
4. W + v = v + W
Associative
Commutative
Identity
Addition & Subtraction
In mathematics, subtraction is an operation that
undoes addition. Subtraction is called the
inverse(opposite) operation of addition.
An addition equation can be rewritten as a
subtraction equation:
Example :
5 + 12 = 17 or 12 = 17 - 5 or 5 = 17 - 12
a + b = c or
a = c - b or
b=c-a
1. Which equation to the right is is
equivalent to 5 + 15 = 20 ?
15 = 5 - 20
5 = 20 - 15
20 = 15 - 5
2. Which equation to the right is
is equivalent to 8 + 9 = 17 ?
8 - 17 = 9
8 - 9 = 17
17 - 8 = 9
3. Which equation to the right is
is equivalent to 24 = 17 + 7 ?
17 - 7 = 24
24 - 7 = 17
17 - 24 = 7
4. Which equation to the right is
is equivalent to 55 = 26 + 29 ?
55 - 26 = 29
26 - 29 = 55
26 - 55 = 29
Review

In mathematics, symbols are often
used to represent ideas.
 For
example, (=) means “is equal to”, the
symbol(>) means “is greater than”, and
(<) means “is less than”.
 The symbols , +, - and X or (•) are the
operation symbols that you have used
many times.
Review cont . . .

Letters are used to represent numbers, and
these letters are referred to as variables.


For example, in the equation 3 + x = 5, x is the letter
that represents the numeral 2.
However, in 3 + x =
, the x is a variable that could
represent many different numerals depending on
the number placed in the blank.
 For example:



3 + x = 7 The variable x represents the numeral 4.
3 + x = 9 The variable x represents the numeral 6.
3 + x = 50 The variable x represents the numeral 47.
Review cont . . .
3+x=7
3+x=9
3 + x = 50


The variable x represents the numeral 4.
The variable x represents the numeral 6.
The variable x represents the numeral 47.
In each equation above, the variable is x , and the
number 3 is called a constant because 3 represents
the same value in each equation.
The answer following the equal sign depends on the
number assigned to the variable (x).
Variables & Multiplication


In multiplication problems, the symbols (X) or
(•) are used to indicate multiplication.
When variables are used, the signs for
multiplication are left out.


For example, “3” multiplied by the variable b will
usually be written as 3b rather than 3 X b or 3•b.
In the expression 3b, the number “3” is the constant
(also known as the coefficient) and the letter b is the
variable.
Variables & Multiplication


The letter a is a variable that could stand for
any number.
Lets work out a multiplication problem involving
a variable.

5a = 30
a=6
Think to yourself: “5 multiplied by
what number is equal to 30?
5 multiplied by 6 equals 30.
Therefore the variable a equals 6.
Variables & Division


When variables are used in division problems,
you will see the problem “8 divided by m as”
8  m or 8/m.
Lets work out a Division problem involving a
variable.

8m=4
m=2
Think to yourself: “8 divided by
what number is equal to 4?
8 divided by 2 equals 4 - - since 4
times 2 equals 8.
Therefore the variable m equals 2
Variable Expressions

Now we will take a look at how variables are
used in algebraic expressions
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