1-4
Simplifying Algebraic
Expressions
Why learn this?
You can model the total points scored
in a basketball game by using an
algebraic expression.
Objective
Simplify and evaluate
algebraic expressions.
There are three different ways in which a
basketball player can score points during
a game. There are 1-point free throws,
2-point field goals, and 3-point field goals.
An algebraic expression can represent the
total points scored during a game.
Total Points Scored
Number of 1-point Number of 2-point
free throws
field goals
Number of 3-point
field goals
To translate a real-world situation into an algebraic expression, you must
first determine the action being described. Then choose the operation that is
indicated by the type of action and the context clues.
Action
Operation
Combine
Add
1
How many total?
Combine equal groups
Multiply
How many altogether?
Separate
Subtract
How many more? How many remaining?
Separate into equal groups
EXAMPLE
Possible Context Clues
Divide
How many in each group?
Translating Words into Algebraic Expressions
Write an algebraic expression to represent each situation.
A the distance remaining for a runner after m miles of a 26.2-mile
marathon
26.2 - m
Subtract m from 26.2.
B the number of hours it takes to fly 1800 miles at an average rate of
n miles per hour
1800
_
n
Divide 1800 by n.
Write an algebraic expression to represent each situation.
1a. Lucy’s age y years after her 18th birthday
1b. The number of seconds in h hours
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To evaluate an algebraic expression,
substitute a number for each variable
and simplify by using the order of
operations. One way to remember
the order of operations is by using
the mnemonic PEMDAS.
Order of Operations
1. Parentheses and grouping symbols
2. Exponents
3. Multiply and Divide from left to right.
4. Add and Subtract from left to right.
EXAMPLE
2
Evaluating Algebraic Expressions
Evaluate each expression for the given values of the variables.
A x + 3xy - 2y for x = 4 and y = 7
(4) + 3 (4) ( 7) - 2 (7)
Substitute 4 for x and 7 for y.
4 + 84 - 14
74
Multiply from left to right.
Add and subtract from left to right.
B b 2z - 2bz + z 2 for b = 6 and z = 2
(6) 2 (2) - 2 (6) (2) + (2) 2
Substitute 6 for b and 2 for z.
36(2) - 2(6)(2) + 4
Evaluate exponential expressions.
72 - 24 + 4
52
Multiply from left to right.
Add and subtract from left to right.
2. Evaluate x 2y - xy 2 + 3y for x = 2 and y = 5.
Recall that the terms of an algebraic expression are separated by addition or
subtraction symbols. Like terms have the same variables raised to the same
exponents. Constant terms are like terms that always have the same value.
Like terms
Constant terms
To simplify an algebraic expression, combine like terms by adding or subtracting
their coefficients. Algebraic expressions are equivalent if they contain exactly the
same terms when simplified.
EXAMPLE
3
Simplifying Expressions
Simplify each expression.
A
Terms that are
written without
a coefficient have
an understood
coefficient of 1.
x 2 = 1x 2
x 2 + 5x + 2y + 7x 2
x 2 + 5x + 2y + 7x 2
8x 2 + 5x + 2y
Identify like terms.
Combine like terms. 1x 2 + 7x 2 = 8x 2
B b(5a 2 - 2a) - 11a 2b + 2ab
5a 2b - 2ab - 11a 2b + 2ab
-6a 2b
Distribute, and identify like terms.
Combine like terms. -2ab + 2ab = 0
3. Simplify the expression -3(2x - xy + 3y) - 11xy.
28
Chapter 1 Foundations for Functions
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Checking Simplified Expressions
To check that I simplified an expression correctly, I substitute the same numbers
into both expressions. If I get the same value for each expression, my answer is
probably correct.
Original Expression
3x + 5y - 2x
3(2) + 5(3) - 2(2)
6 + 15 - 4
17
Nadia Torres
Madison High School
EXAMPLE
4
Simplified Expression
Use x = 2 and y = 3.
Multiply.
They are equal.
x + 5y
2 + 5(3)
2 + 15
17
Transportation Application
Holly’s hybrid car gets 45 miles per gallon on
the highway and 25 miles per gallon in the city.
A Write and simplify an expression for the
total number of miles she can drive if
her fuel tank holds 15 gallons of gas.
Let h be the number of gallons used on
the highway. Then 15 - h is the remaining
number of gallons used in the city.
45h + 25(15 - h) = 45h + 375 - 25h Distribute 25.
Combine like terms.
= 20h + 375
B How many total miles can she drive on one tank of gas if she
uses 5 gallons on the highway?
Evaluate 20h + 375 for h = 5.
20 (5) + 375 = 475
Holly can travel 475 miles if she uses 5 gallons on the highway.
4. A travel agent is selling 100 discount packages. He makes
$50 for each Hawaii package and $80 for each Cancún package.
a. Write an expression to represent the total the agent will
make selling a combination of the two packages.
b. How much will he make if he sells 28 Hawaii packages?
THINK AND DISCUSS
1. Tell how many addition or subtraction symbols an expression with five
terms will have. Explain.
2. Explain how adding like terms involves
the Distributive Property.
3. GET ORGANIZED Copy and complete
the graphic organizer. In each box,
write key words that may indicate
each operation.
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