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 : 

3 ⋅ 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 : 

m ÷ 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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