# VARIABLES AND EXPRESSIONS

Many recycled products are used in a "green built" house, including carpet made from recycled plastic drink bottles. We will be able to determine how many square feet of carpet can be made from a certain number of plastic drink bottles by using variables, constants and expression.

A variable is a letter or symbol used to represent a value that can change.

A constant is a value that does does not change.

A numerical expression may contain only constants and/or operations.

A algebraic expression may contain variables, constants and/or operations.

To be successful in Math, we may have to translate between algebraic expressions and words.

## Translating from Algebraic Symbols to Words

Let us see some of the ways to write mathematical operations with words.

Plus, sum, increased by -----> +

Minus, difference, less than -----> -

Times, product, equal groups of -----> x

Divided by, quotient -----> ÷

Give two ways to write each algebraic expression in words.

Example 1 :

y + 7

Solution :

(i) The sum of y and 7

(ii) y increased by 7

Example 2 :

⋅ p

Solution :

(i) 3 times p

(ii) The product of 3 and p

Example 3 :

k - 5

Solution :

(i) The difference of k and 5

(ii) 5 less than k

Example 4 :

÷ 2

Solution :

(i) m divided by 2

(ii) The quotient of m and 5

## Translating from Words to Algebraic Symbols

To translate words into algebraic expressions, look for words that indicate the action that is taking place.

Put together, combine -----> Add

Find how much more or less -----> Subtract

Put together equal groups -----> Multiply

Separate into equal groups -----> Divide

Example 5 :

Jacob reads 18 pages per hour. Write an expression for the number of pages Jacob reads in x hours.

Solution :

Think : x groups of 18 pages.

18 ⋅ x  or  18x

Example 6 :

John is 3 years younger than Michael, who is m years old.  Write an expression for John's age.

Solution :

m represents Michael's age

Think : "younger than" means " less than".

John's age :

m - 2

Example 7 :

Neil runs a mile in 14 minutes. Write an expression for the number of miles that Neil runs in y minutes.

Solution :

y represents the total time Neil runs

Think : How many groups of 14 are in y ?

y/14

Example 8 :

Lily is 7 cm taller than her sister, who is k cm tall. Write an expression for Lily's height in centimeters.

Solution :

k represents the height of Lily's sister

Lily's height in centimeters :

k + 7

## Evaluating Algebraic Expressions

Example 9 :

Evaluate the following expression for x = 2 and y = 3.

x + y

Solution :

=  x + y

Substitute 2 for x and 3 for y.

=  2 + 3

=  5

Example 10 :

Evaluate the following expression for y = 24 and z = 4.

y/z

Solution :

=  y/z

Substitute 24 for y and 4 for z.

=  24/4

=  6

Example 11 :

Evaluate the following expression for x = 3 and y = 4.

x2y + y

Solution :

=  x2y + y

Substitute 3 for x and 4 for y.

=  32(4) + 4

Use PEMDAS.

=  9(4) + 4

=  36 + 4

=  40

Example 12 :

Evaluate the following expression for x = 5 and y = 3.

(x + 2y)2 - 2x

Solution :

=  (x + 2y)2 - 2x

Substitute 5 for x and 3 for y.

(5 + 2 ⋅ 3)2 - 2 ⋅ 5

Use PEMDAS.

=  (5 + 6)2 - 10

=  112 - 10

=  121 - 10

=  111

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