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2.14: Variables

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    Learning Objectives

    • Define and identify variables

    Two men dressed similarly, with the same haircut, and both wear round glasses take a selfie with each other.Greg and Alex have the same birthday, but they were born in different years. This year Greg is 20 years old and Alex is 23, so Alex is 3 years older than Greg. When Greg was 12, Alex was 15. When Greg is 35, Alex will be 38. No matter what Greg’s age is, Alex’s age will always be 3 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 3 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 g. Then we could use g+3 to represent Alex’s age. See the table below.

    Greg’s age Alex’s age
    12 15
    20 23
    35 38
    g g+3

    Letters are used to represent variables. Letters often used for variables are x,y,a,b,\text{ and }c.

    Variables and Constants

    A variable is a letter that represents a number or quantity whose value may change (ex. x, y, z, a, t, k etc.).

    A constant is a number whose value always stays the same.


    Identify the variable(s) in each expression or equation

    1. x+2
    2. 5-3y
    3. 7+5b-z=9

    [reveal-answer q=”114888″]Show Answer[/reveal-answer]
    [hidden-answer a=”114888″]


    1. x
    2. y
    3. b and z


    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 a+b a\text{ plus }b the sum of a and b
    Subtraction a-b a\text{ minus }b the difference of a and b
    Multiplication a\cdot b,\left(a\right)\left(b\right),\left(a\right)b,a\left(b\right) a\text{ times }b The product of a and b
    Division a\div b,a/b,\frac{a}{b},b\overline{)a} a divided by b The quotient of a and b

    In algebra, the cross symbol, \times, is not used to show multiplication because that symbol may cause confusion. Does 3xy mean 3\times y (three times y ) or 3\cdot x\cdot y (three times x\text{ times }y )? 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.

    \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}


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