Math & History: Variables & Expressions

I. Intro

A variable is a symbol or (more often) a letter used to represent one or more numbers. The numbers are called the values of the variable. At an elementary level variables represent an unknown amount.

Example:

A constant is a fixed numerical value.

Example:

A coefficient is a number multiplied by a variable.

Example:

14c

Here, 14 is the coefficient to be multiplied by variable c.

A factor is one part of a product. Factors are the numbers and variables that are multiplied together.

Example:

In the term 8x, the factors are 8 and x.

A term is either a single number or variable, or the product of numbers or variables.

Example:

An expression is a combination of terms that are combined by using the mathematical operations addition and subtraction. It can be thought of as a mathematical sentence. Expressions never have an equal sign.

Example:

g + 25 - 7

An equation is a mathematical sentence that says two expressions are equal. Equations always have an equal sign.

Example:

g + 25 - 7 = 7g

The Cross Symbol for Multiplication

In algebra, the cross symbol, X, is not used to show multiplication since the symbol can be confused as a variable

The factors of the term 5x are 5 & x

II. Evaluating Expressions

To evaluate an algebraic expression means to find the value of the expression when the variable is replaced by a given number. To evaluate an expression, we substitute the given number for the variable in the expression and then simplify the expression using the order of operations.

Example (1 variable):

Evaluate 9x - 2, when x = 5

9x - 2

= 9(5) - 2

= 45 - 2

= 43

Example (2 variables):

10 + 2p - 3r, when p = 4 & r = 5

= 10 + 2(4) - 3(5)

= 10 + 8 - 15

= 3

III. Simplify Expressions Combining Like Terms

We can simplify an expression by combining the like terms. Like terms are terms where the variables match exactly (exponents included). To do this, add the coefficients and keep the same variable.

Example:

3x + 6x = 9x

We can see why this works by writing both terms as addition problems.

Add the coefficients and keep the same variable. It doesn’t matter what x is. If you have 3 of something and add 6 more of the same thing, the result is 9 of them. For example, 3 oranges plus 6 oranges is 9 oranges.

Example: (Fraction)

Khan: Combining like terms with negative coefficients

IV. Distributive Property

The distributive property states that multiplying a number by the sum or difference of two numbers is the same as multiplying the number by each of the numbers individually and then adding or subtracting the results

Stated with variables, the distributive property of multiplication over addition states:

x(y + z) = x(y) + x(z)

We can also use the distributive property of multiplication over subtractions:

x(y - z) = x(y) - x(z)

The Distributive Property with Variables

The distributive property is useful when simplifying expressions with variables. When using variables, you may not be able to do the operation inside the parentheses first.

Example:

8(y + 4z)

= 8(y) + 8(4z)

= 8y + 32z

Example: Distributing a negative sign

5x - (3x + 2)

=5x -3x - 2

Khan: Combining like terms with negative coefficients and distribution

Factoring with the distributive property

V. Writing Expressions Word Problems

Khan: Writing Expressions Word Problems

VI. Miscellaneous

Variable In Numerator

When writing a term that has a fraction and a variable in the numerator, sometimes the variable is written in the numerator, sometimes it is written on the side. Both have the same meaning.

Dividing An Expression (Polynomial) By A Number

Dividing an expression by a number involves dividing each term individually by that number.

Example: Divide 6x² + 9x + 12 by 3