Variables

Tutorial 3c

A variablevariable 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

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).

Example:

x

2

6

6 + 2 = x x must be 8 for this open sentence to be true

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.

Variables

Examples6 + 2 = x x must be 8 for this equation to

be true

y = 12 - 6 y must be 6

1. A = 6 + 2 A = 8 104

2. 23 - 6 = x x = 178 13

3. G = 12 + 7 G = 195 31

4. 21 - 7 = d d = 14 17 28

Find the value of the variable that makes each equation true.

Addition PropertiesAddition PropertiesCommutative Property of Addition :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 PropertiesAddition PropertiesAssociative Property of Addition: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 PropertiesAddition Properties

Additive Identity of Addition: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

1. 9 + 0 = 9CommutativeAssociative Identity

2. 7 + 8 = 8 + 7CommutativeAssociative Identity

3. (2 + 4) + 3 = 3 + (2 + 4)CommutativeAssociative Identity

4. W + v = v + WCommutativeAssociative Identity

Choose the correct property.

Addition & SubtractionAddition & Subtraction

In mathematics, subtraction is an operation that In mathematics, subtraction is an operation that undoes addition. Subtraction is called the undoes addition. Subtraction is called the inverse(opposite) operation of addition. inverse(opposite) operation of addition.

Example : 5 + 12 = 17 or 12 = 17 - 5 or 5 = 17 - 12

a + b = c or a = c - b or b = c - a

An addition equation can be rewritten as a An addition equation can be rewritten as a subtraction equation:subtraction equation:

1. Which equation to the right is is equivalent to 5 + 15 = 20 ? 5 = 20 - 15

20 = 15 - 5

15 = 5 - 20

2. Which equation to the right is is equivalent to 8 + 9 = 17 ?

17 - 8 = 9

8 - 9 = 17

8 - 17 = 9

3. Which equation to the right is is equivalent to 24 = 17 + 7 ? 24 - 7 = 17

17 - 24 = 7

17 - 7 = 24

4. Which equation to the right is is equivalent to 55 = 26 + 29 ?

55 - 26 = 29

26 - 55 = 29

26 - 29 = 55

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 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 & 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 Think to yourself: “5 multiplied by

what number is equal to 30?

5 multiplied by 6 equals 30.

Therefore the variable a equals 6.

a = 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 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

m = 2

Variable Expressions

Now we will take a look at how variables are used in algebraic expressions

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