Wednesday, August 11, 2010

Algebraic Expressions

Algebraic Expressions Introduction

An algebraic expression is a combination of numbers, letters, and four basic arithmetic operations.

The numbers in an algebraic expression are called constants.

The letters in an algebraic expression are called variables. In an algebraic expression, letters always stand for numbers.

The following are some examples of algebraic expressions.

n – 2
Y / 3
11x

a + 5

Algebraic expressions may contain more than one number, more than one variable, more than one arithmetic operation.

For example:

3x – 7y + 11
5n × 9p ÷ 6

Algebraic expressions often include parentheses (brackets) and exponents.

For example:

3(a – 4)
(y – 4)(y + 8)
5(m^3)
9(a^2) – 5(b^2) + 131

Algebraic expressions do not contain a relation symbol like <, >, <=, >= etc.

Terms in an algebraic expression

A term in an algebraic expression is a constant or a variable or both multiplied together.

For example:

3x – 16y + 11 has three terms 3x, –16y, and 11
3x and –16y are called the ‘variable terms’ and 11 are called the ‘constant term’.

Every term in an algebraic expression has a coefficient. To learn more about coefficients, click the link below or copy/paste the link below.

http://www.icoachmath.com/SiteMap/Coefficient.html

Evaluating algebraic expressions

Algebraic expressions may be evaluated by substituting the given values for the variables in the expression.

http://www.icoachmath.com/SiteMap/EvaluateanAlgebraicExpression.html

Solved Examples on Algebraic Expression

Example 1

Write an algebraic expression for the statement below.
Seven times a certain number

Solution:

We can use any letter to represent ‘the number’. Let’s use ‘n’.

So:

Seven times a certain number = 7 × a certain number = 7 × n = 7n

Example 2

Write the following phrase as an algebraic expression.

Nine more than thrice a number

Solution:

We can use any letter to represent ‘the number’. Let’s use ‘m’ here.

Before we proceed with the conversion, let’s first understand the meaning of the key words in the given phrase.

The key word ‘more than’ stands for the operation ‘addition’.
The key word "thrice" means ‘three times’.

So,

Thrice a number = 3 × a certain number = 3 × m = 3m
Therefore,

Nine more than thrice a number = 3m + 9

We can represent real-life situations using algebraic expressions.

Example 1

Mathew and his three college friends are going to be sharing the cost of a 3 bedroom apartment. The cost of rent is \$n. Write an expression that tells us what Mathew's share is.

Solution:

The cost of rent is \$n.

This cost is going to be shared by 4 people >> Mathew + his 3 friends.

This means:

The cost \$n is going to be EQUALLY DIVIDED among the 4 people.

So, each one’s share would be \$n/4.

Example 2

A cassette shop charges \$5 for renting a video cassette for the first day. It charges \$2 extra for every additional day. Which of these algebraic expressions tells you the amount that Sarah would pay if she rents a video cassette for n days?

Choices:
A. \$(3 + 2n)
B. \$(5n + 2)
C. \$ n
D. \$(5 - 2n)

Solution:

Step 1: \$5 for the first day and \$2 extra for every additional day.

Step 2: The amount that Sarah would pay if she rents a video cassette for n days = Rent for the first day + Rent for the remaining days.

Step 3: = \$5 + Remaining day’s × \$2

Step 4: = \$5 + (n – 1) × \$2 [Remaining days = n days – first day = n – 1.]

Step 5: So, Sarah would pay \$5 + (n - 1) × \$2 for renting a video cassette for n days.

Step 6: Therefore, the algebraic expression that corresponds to the amount that Sarah would pay is \$(3 + 2n).

Example 3

Write a real life situation that matches the following algebraic expression.

X ÷ 14

Solution:

A certain number (x) is divided by the number 14.

The given algebraic expression can describe the following situation.

Amy distributed ‘x’ cookies among 14 friends. The algebraic expression x ÷ 14 represents the number of cookies each friend gets.

Try and come up with your own examples.

Example 4

Write a real life situation that matches the following algebraic expression.

7 + m

Solution:

A certain number (m) is added to 7.

The given algebraic expression can describe the following situation.

Lisa has 7 marbles. Her brother gives her some more. The algebraic expression 7 + m represent the number of marbles that Lisa has in all.

Try and come up with your own examples.
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Math Dictionary