Learning Objectives
- Define and identify variables
Greg and Alex have the same birthday, but they were born in different years. This year Greg is [latex]20[/latex] years old and Alex is [latex]23[/latex], so Alex is [latex]3[/latex] years older than Greg. When Greg was [latex]12[/latex], Alex was [latex]15[/latex]. When Greg is [latex]35[/latex], Alex will be [latex]38[/latex]. No matter what Greg’s age is, Alex’s age will always be [latex]3[/latex] years more, right?
In the language of algebra, we say that Greg’s age and Alex’s age are variable and the three is a constant. The ages change, or vary, so age is a variable. The [latex]3[/latex] years between them always stays the same, so the age difference is the constant.
In mathematics, letters of the alphabet are used to represent variables. Suppose we call Greg’s age [latex]g[/latex]. Then we could use [latex]g+3[/latex] to represent Alex’s age. See the table below.
Greg’s age | Alex’s age |
---|---|
[latex]12[/latex] | [latex]15[/latex] |
[latex]20[/latex] | [latex]23[/latex] |
[latex]35[/latex] | [latex]38[/latex] |
[latex]g[/latex] | [latex]g+3[/latex] |
Letters are used to represent variables. Letters often used for variables are [latex]x,y,a,b,\text{ and }c[/latex].
Variables and Constants
A variable is a letter that represents a number or quantity whose value may change (ex. [latex]x, y, z, a, t, k[/latex] etc.).
A constant is a number whose value always stays the same.
EXAMPLE
Identify the variable(s) in each expression or equation
- [latex]x+2[/latex]
- [latex]5-3y[/latex]
- [latex]7+5b-z=9[/latex]
TRY IT
To write algebraically, we need some symbols as well as numbers and variables. There are several types of symbols we will be using. There are multiple symbols and phrases to represent the four basic arithmetic operations: addition, subtraction, multiplication, and division. We will summarize them here:
Operation | Notation | Say: | The result is… |
---|---|---|---|
Addition | [latex]a+b[/latex] | [latex]a\text{ plus }b[/latex] | the sum of [latex]a[/latex] and [latex]b[/latex] |
Subtraction | [latex]a-b[/latex] | [latex]a\text{ minus }b[/latex] | the difference of [latex]a[/latex] and [latex]b[/latex] |
Multiplication | [latex]a\cdot b,\left(a\right)\left(b\right),\left(a\right)b,a\left(b\right)[/latex] | [latex]a\text{ times }b[/latex] | The product of [latex]a[/latex] and [latex]b[/latex] |
Division | [latex]a\div b,a/b,\frac{a}{b},b\overline{)a}[/latex] | [latex]a[/latex] divided by [latex]b[/latex] | The quotient of [latex]a[/latex] and [latex]b[/latex] |
In algebra, the cross symbol, [latex]\times [/latex], is not used to show multiplication because that symbol may cause confusion. Does [latex]3xy[/latex] mean [latex]3\times y[/latex] (three times [latex]y[/latex] ) or [latex]3\cdot x\cdot y[/latex] (three times [latex]x\text{ times }y[/latex] )? To make it clear, use • or parentheses for multiplication.
Grouping symbols in algebra are much like the commas, colons, and other punctuation marks in written language. They indicate which expressions are to be kept together and separate from other expressions. The table below lists three of the most commonly used grouping symbols in algebra.
Common Grouping Symbols | |
---|---|
parentheses | ( ) |
brackets | [ ] |
braces | { } |
Here are some examples of expressions that include grouping symbols.
[latex]\begin{array}{cc}8\left(14 - 8\right)21 - 3\\\left[2+4\left(9 - 8\right)\right]\\24\div \left\{13 - 2\left[1\left(6 - 5\right)+4\right]\right\}\end{array}[/latex]