VARIABLES AND EXPRESSIONS

A variable is one of the twenty six English alphabets or any symbol like θ used to represent a value that can change.

A constant is a number that does not change.

A numerical expression is the one which contains only the numbers and one or more binary operations. An algebraic expression is the one which contains variables, constants, and one or more more binary operations. To be successful in mat, we may have to do translation between algebraic expressions and words. 

Four binary operations in math with words. 

+ ----> plus or sum or increased by

- ----> minus or difference or less than

x ----> times or product or equal groups of

÷ ---> divided by or quotient

Translating from Algebraic Symbols to Words

Write each algebraic expression in words in two ways.

Example 1 :

x + 5

Answer :

• The sum of x and 5.
• x is increased by 5.

Example 2 :

y - 9

Answer :

• The difference of y and 9.
• 9 less than y.

Example 3 :

2 · m

Answer :

2 times m.

The product of 2 and m.

Example 4 :

a ÷ 6

Answer :

• a divided by 6.
• The quotient of a and 6.

To do translation between words and algebraic expressions, look for words that indicate the action that is taking place.

Add ----> put together or combine

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

Multiply ----> Put together equal groups

Divide ---> Separate into equal groups

Translating from Words to Algebraic Symbols

Example 5 :

Thomas reads 22 pages per hour. Write an expression for the number of pages he reads in m hours.

Answer :

m represents the number of hours that Thomas reads.

Think : m groups of 25 pages.

25 · m or 25m

Example 6 :

John is 2 years younger than Kay, who is k years old. Write an expression for John’s age.

Answer :

k represents Kay's age.

Think : 'younger than' means 'less than'.

k - 2

Example 7 :

Peter runs a mile in 15 minutes. Write an expression for the number of miles that Peter runs in y minutes.

Answer :

y represents the total time Peter runs.

Think : How many groups of 15 are in y.

y ÷ 15 or y/15

Evaluating Algebraic Expressions

Evaluate each expression for x = 7, y = 4, and z = 2.

Example 8 :

x + y

Answer :

x + y = 7 + 4

= 11

Example 9 :

y/z

Answer :

y/z = 4/2

= 2

Example 10 :

xyz

Answer :

xyz = 7 · 4 · 2

= 56

Application

Example 11 :

Approximately ten 25-ounce plastic drink bottles must be recycled to produce 1 square foot of carpet.

a. Write an expression for the number of bottles needed to make x square feet of carpet.

b. Find the number of bottles needed to make 20, 50, and 200 square feet of carpet.

Answer :

a. The expression 10x models the number of bottles needed to make x square feet of carpet.

b. Evaluate 10x for x = 20, 50, and 200.

x = 20 ----> 10(20) = 200

x = 50 ----> 10(50) = 500

x = 200 ----> 10(200) = 2000

To make 20 ft² of carpet, 200 bottles are needed.

To make 50 ft² of carpet, 500 bottles are needed.

To make 200 ft² of carpet, 2000 bottles are needed.

Example 12 :

The length of Interstate 90 from the West Coast to the East Coast is 153.5 miles more than 2 times the length of Interstate 15 from southern California to northern Montana. Let m be the length of Interstate 15.

Which expression can you use to represent the length of Interstate 90?

a) 2m + 153.5     b)  2m − 153.5      c) 153.5 − 2m      d)  153.5m − 2

Answer :

Length of interstate 90 from West Coast to the East Coast = 153.5

Two times, then 2m

More than, so use +. Then the answer is 2m + 153.5, option a is correct.

Example 13 :

You plant a cypress tree that is 10 inches tall. Each year, its height increases by 15 inches.

a. Make a table that shows the height of the tree for 4 years. Then write an expression for the height after t years.

b. What is the height after 9 years?

Answer :

The initial height of cypress tree = 10 inches

Every year, its height increases by 15 inches. Then + 15 

Let t be the number of years. Then the required expression is 10 + 15t

So, the height after year t is 10 + 15t.

b. Evaluate 10 + 15t

when t = 9.

10 + 15t = 10 + 15(9) = 145

After 9 years, the height of the tree is 145 inches