variables & algebraic expressions variable algebraic expression a symbol, usually a letter, that...
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Variables & Algebraic Expressions
Variable
Algebraic Expression
A symbol, usually a letter, that represents one or more numbers.
A mathematical phrase that includes one or more variables.
hoursworke
d
pay(in
dollars)
1 60
2 120
3 180
4 240
5 300
6 360
If a plumber earns $60 per hour, in order to calculate how much money he made for a particular job, you would multiply the number of hours he worked by $60.You could model this relationship with an algebraic expression using h as a variable to represent the number of hours worked.
60h
Numerical Expression
A mathematical phrase involving numbers and operation symbols, but no variables.
Equation
A mathematical sentence that has an equal sign.
Verbal Phrases Involving AdditionThe algebraic expression a + b can be used to represent several different verbal phrases.
a plus b a added to b
the sum of a and b b is added to a
a is increased by b
b more than a
The word exceeds means “is more than”. So, the number that exceeds 5 by 2 can be written as “2 more than 5” or 5 + 2.
Verbal Phrases Involving Subtraction
The algebraic expression a - b can be used to represent several different verbal phrases.
a minus b a decreased by b
b subtracted from a a diminished by b
b less than a
a reduced by b
the difference between a and b
Verbal Phrases Involving Multiplication
The algebraic expressions a X
b, a . b, (a)(b), and ab can be used to represent several different verbal phrases.
a times b the product of a and b b multiplied by a
The preferred form to indicate multiplication in algebra is ab.
The multiplication symbol X is avoided in algebra because it can be confused with the variable x.
In algebraic expressions, parenthesis may be used but are not needed.( ) 3)(3 b ba a
Verbal Phrases Involving DivisionThe algebraic expressions
a divided by b the quotient of a and b
a b and
ab
can be used to represent several different verbal phrases.
4a and 4a
means one-forth of a as well as a divided by 4. Commas are
very important in some verbal
phrases..
The product of x and y, decreased by 2.
2xy
The product of x and y decreased by 2. ( )2x y
Translating Words to Algebra
3 more than a number 3n
5 less than a number 5n
a number decreased by 7 n 7
the product of 6 and a number 6n
twice a number decreased by 4 2n 4
25, diminished by 4 times a number 25 4n
the sum of a number and 2, divided by 92n
9
These are
examples
of Algebraic
Expression
s.
Got It?1)
What is an algebraic expression for each word phrase?
What is an algebraic expression for 18 more than a number n?
2)
6 times a number na)
The quotient of 18 and a number nb)
What is an algebraic expression for each word phrase?3)
8 less than the product of a number x and 4a)
Twice the sum of a number x and 8b)
The quotient of 5 and the sum of 12 and a number x
c)
What word phrase can you use to represent the algebraic expression?
4)
a) 8.1x b)10 9x c)3n d) 5 1x
18n
6n 18n
4 8x 2( 8)x
512 x
The sum of a number x and 8.1
The sum of 10 times a number x and 9
The quotient of a number n and 3
Five times a number x less 1
Homework
Page 7: 10 - 24 even #’s only
Modeling Relationships with Equations
Equation
Open Sentence
A mathematical sentence that uses an equal sign.An equation that contains one or more variables.
Brianna paid 17 dollars for batteries and film for her camera. If the batteries cost x dollars, express the cost of the film in terms of x.
Relate: Film equals $17 minus the cost of the batteries.
Define:x = cost of batteriesf = cost of film
Write: 17f x
That was easy
More Modeling Relationships with Equations
CD’sCost(in
dollars)
1 15
2 30
3 45
4 60
Define the variable and write an equation to model the relationship in the table.
Define:n = number of CD’sc = cost of CD’s
Write:
15c n
Number of Workers
Radios Built
1 11
2 22
3 33
4 44
Define:w = number of workersr = number of radios
Write:
11r w
Asi de
Facil
Even More Modeling Relationships with Equations
SalesEarnings
(in dollars)
3 21
6 42
9 63
12 84
Define the variable and write an equation to model the relationship in the table.
Define:n = number of salese = earningsWrite:
7e n
Number of Hours
Total Pay
5 $37.50
10 $75.00
15 $112.50
20 $150.00
Define:n = number of hoursp = total payWrite:
7.50p n
Asi de
Facil
First divide to determine the earnings for one sale.
213
7First divide to determine the pay for one hour.
537.50
7.50
Writing a Rule to Describe a Pattern
A state park charges an entrance fee of $ 20, plus $18 for each night of camping. a) If n is the number of nights that you
spend camping, write an expression that gives the total cost, c, of camping for n nights. 18 20c n
total cost of campin
g
price per
night
# of nights
entrance fee
(constant)
b) What is the total cost of camping for 3 nights?
318 2( 0)c
54 20c $74c That was
easy
Homework
Page 8: 25, 26, 32, 33, 34
Order of OperationsLet’s look at an example of a
computation with more than one operation.
11 3 2
8 216
11 65
I’ll try doing 11 – 3 first.
Now I’ll try doing 3 times 2 first.
Please Excuse My Dear Aunt Sally
Please Excuse My Dear Aunt Sally
ParenthesesExponentsMultiplicationDivisionAdditionSubtraction
That was easy
ExamplesSimplify the following. For all fractions, express the exact value and the rational approximation rounded to the nearest hundredth.
2342
323
447
3 2(7 )
26 1 3 4( )
8 2(6 )
212 18( )3
ExamplesSimplify the following. For all fractions, express the exact value and the rational approximation rounded to the nearest hundredth.
2 ( ) 93 3 (3) 4 ( )( )( )(2 2 2 2 2 1) 6
3 3
3
2 2 2 2( ) ( )( )( )( ) (
23 3 3)(3)(
827)3
4 4
4
( ) ( )( )( )( )4 4 4 4 4 47 7 7( ) ( )( )( )(7 7 7)
2562401
ExamplesSimplify the following. For all fractions, express the exact value and the rational approximation rounded to the nearest hundredth.
3 7 2 3) 5( 15
2 26 1 3 4 6 4( ) ( 4 16 4) 6 6
8 6 2 8) 3( 11
2 2( ) (12 18 3 12 6 12 36) 432
Evaluating Algebraic Expressions
To evaluate an algebraic expression, replace the variables with the given values, and then follow the proper order of operations.
Don’t forget about my dear Aunt Sally.
Evaluate 50 3xwhen 7x
Rewrite the expression with empty parenthesis where the variables are
(0 )5 3
Replace the variables with the given values
50 )3(7
Follow the proper order of operations 50 21 29
More Evaluating Algebraic Expressions
Evaluate 22 5 4x x when 7x
22 5( ) ( 4 ) 7 7
2 5( ) ( )49 47 98 435
137
1.5x 22 5( ) ( 4 ) 1.5 1.5
2 5( ) (2.25 1 4.5) 4.5 7 4.5
1
Evaluate2
)15
(n da
when 4a 10n 3d
( )( ) (
21
5 )
410 3
9 35
( )8
)(
8 275 1
55
8 275 1
8 1355 5
1275
25.4 or 2
255
when
Using a Calculator to Evaluate Algebraic Expressions
Evaluate 2)1
5(n d
a
when 4a 10n 3d
The given values can be stored in the calculator.
Press (-) 4 STO ALPHA A ENTER
10 STO ALPHA N ENTER
3 STO ALPHA D ENTER
2A/5+(N-1)D ENTER 25.4
Evaluate2
)15
(n da
when 10a 6n
10 STO ALPHA A ENTER
(-) 6
5d
STO ALPHA N ENTER
(-) 5 STO ALPHA D ENTER
2A/5+(N-1)D ENTER
39
Homework
Page 13: 10 – 34 even #’s only