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1.5.3: Evaluate Expressions with One or More Variables

  • Page ID
    4282
  • Expressions with One or More Variables

    Figure \(\PageIndex{1}\)

    Dexter is in charge of ticket sales at his town’s water park. He has to report to his boss how many tickets he sells and how much money the water park makes in ticket sales each day - adult tickets ($7), child tickets ($5). Today, Dexter has sold 100 adult and 125 child tickets. Yesterday, he sold 120 adult and 120 child tickets. Can Dexter write an expression to figure out today’s ticket sales, and use this to compare today’s sales to yesterday’s?

    In this concept, you will learn how to evaluate expressions that have multiple variables and/or multiple operations.

    Expressions with Multiple Variables

    When evaluating expressions with multiple variables and multiple operations, it is important to remember the order of operations.

    Order of Operations:

    P - parentheses

    E- exponents

    MD - multiplication and division, in order from left to right

    AS - addition and subtraction, in order from left to right

    Whenever you are evaluating an expression with more than one operation in it, always refer back to the order of operations.

    Let's look at an example of an expression with multiple variables and operations.

    Evaluate 6a+b when a is 4 and b is 5.

    First, you can see that there are two variables in this expression, a and b. There are also two operations here: multiplication, seen in "6 times the value of a" and addition, seen in "+b". You are given values for a and b.

    First, substitute the given values for each variable into the expression.

    6a+b

    6(4)+5

    Then, evaluate the expression according to order of operations.

    6(4)+5

    24+5

    29

    The answer is 29.

    Let’s look at another example with multiple variables and expressions.

    Evaluate 7b−d when b is 7 and d is 11.

    First, substitute the given values in for the variables.

    7(7)−11

    Then, evaluate the expression according to order of operations.

    49−11

    38

    The answer is 38.

    Sometimes you may have an expression that is all variables. Evaluate this in the same way.

    Evaluate ab+cd when a is 4, b is 3, c is 10 and d is 6.

    First, substitute the given values in for the variables.

    ab+cd

    (4)(3)+(10)(6)

    Next, evaluate the expression according to order of operations.

    12+60

    The answer is 72.

    Examples

    Example \(\PageIndex{1}\)

    Earlier, you were given a problem about Dexter and his tickets.

    Dexter needs to write an expression to figure out the total money made from the 100 adult and 125 child tickets he sold today, compared to the 120 adult and 120 child tickets he sold yesterday. The adult tickets cost $7 and the child tickets cost $5.

    Solution

    First, write an expression.

    7x+5y

    Next, substitute in the given values for today’s ticket sales.

    7(100)+5(125)

    Then, follow order of operations to multiply and then add.

    7(100)+5(125) Multiply 7×100

    700+5(125) Multiply 5×125

    700+625 Add 700+625=1325

    1,325

    The answer is 1,325.

    Next, substitute in the given values for yesterday’s ticket sales.

    7(x)+5(y)

    7(120)+5(120)Substitute 120 for both x and y

    Then, follow order of operations to multiply and then add.

    7(120)+5(120) Multiply 7×120

    840+5(120) Multiply 5×120

    840+600 Add 840+600=1440

    1,440

    The answer is 1,440.

    Finally, subtract 1,325 from 1,440 to get the difference.

    1,440−1,325=115

    The answer is 115.

    Dexter can report to his boss that today the park sold $1,325 in tickets, which is $115 less than the $1,440 they sold in tickets yesterday.

    Example \(\PageIndex{1}\)

    Evaluate a+ab+cd when a is 4, b is 9, c is 6 and d is 4.

    Solution

    First, substitute the given values into the expression.

    4+4(9)+6(4)

    Next, evaluate according to order of operations.

    4+36+24

    64

    Example \(\PageIndex{1}\)

    Evaluate 12x−y when x is 4 and y is 9.

    Solution

    First, substitute 4 for x and 9 for y.

    12x−y Substitute 4 in place of x and 9 for y

    12(4)−(9)

    Then, evaluate the expression.

    12(4)−(9) Multiply 12×4=48

    48−(9) Subtract 48−9=39

    39

    Example \(\PageIndex{1}\)

    Evaluate (12/a)+4 when a is 3.

    Solution

    First, substitute 3 for a.

    5x+3y

    Next, evaluate the expression.

    123+4 Divide 12÷3=4

    4+4 Add 4+4=8

    8

    Example \(\PageIndex{1}\)

    Evaluate 5x+3y when x is 4 and y is 8.

    Solution

    First, substitute 4 for x and 8 for y.

    5x+3y Substitute 4 for x and 8 for y

    5(4)+3(8)

    Then, evaluate the expression.

    5(4)+3(8) Multiply 5×4=20

    20+3(8) Multiply 3×8=24

    20+24 Add 20+24=44

    44

    The answer is 44.

    Review

    Evaluate each multi-variable expression when x=2 and y=3.

    1. 2x+y
    2. 9x−y
    3. x+y
    4. xy
    5. xy+3
    6. 9y−5
    7. 10x−2y
    8. 3x+6y
    9. 2x+2y
    10. 7x−3y
    11. 3y−2
    12. 10x−8
    13. 12x−3y
    14. 9x+7y
    15. 11x−7y

    Review (Answers)

    To see the Review answers, open this PDF file and look for section 1.14.

    Vocabulary

    Term Definition
    algebraic The word algebraic indicates that a given expression or equation includes variables.
    Algebraic Expression An expression that has numbers, operations and variables, but no equals sign.
    Exponent Exponents are used to describe the number of times that a term is multiplied by itself.
    Expression An expression is a mathematical phrase containing variables, operations and/or numbers. Expressions do not include comparative operators such as equal signs or inequality symbols.
    Order of Operations The order of operations specifies the order in which to perform each of multiple operations in an expression or equation. The order of operations is: P - parentheses, E - exponents, M/D - multiplication and division in order from left to right, A/S - addition and subtraction in order from left to right.
    Parentheses Parentheses "(" and ")" are used in algebraic expressions as grouping symbols.
    revenue Revenue is money that is earned.
    substitute In algebra, to substitute means to replace a variable or term with a specific value.
    Variable Expression A variable expression is a mathematical phrase that contains at least one variable or unknown quantity.

    Additional Resources

    Videos:

    PLIX: Play, Learn, Interact, eXplore: Expressions with One or More Variables: Water Bottle Expression

    Activity: Expressions with One or More Variables Discussion Questions

    Practice: Evaluate Expressions with One or More Variables

    Real World Application: MotoGP