Download Variable Expressions

Digital Lesson
Variable Expressions
These are examples of variable expressions.
a  3b
2
1
x  5
2
2(4 x  1)
 9 xy 


 6 


3

A variable (or algebraic) expression is an expression
formed from numbers and variables by adding,
subtracting, multiplying, dividing, taking powers,
taking roots, and using grouping symbols.
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
2
Replacing the variables in a variable expression by numbers
produces a numerical expression. When this is evaluated the
resulting number is the value of the variable expression.
Examples: 1. Find the value of 3x – 5 when x = – 1.
= 3(– 1) – 5 Replace the variable x
with the number – 1.
= –3 – 5
Value
= –8
4x  2
2. Find the value of
when x = 4.
9
Replace the variable x
4(4)  2

with the number 4.
9
16  2 18
Value

 2
9
9
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
3
We use variable expressions to represent verbal expressions.
Examples of verbal expressions:
These can be translated into
variable expressions:
“3 years older than Alice”
a+3
a = Alice’s age
“4 pizzas less than we served yesterday”
p–4
p = number of pizzas
served yesterday
“8 times as many nickels as quarters”
n = 8q
q = number of quarters
n = number of nickels
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
4
Examples: 1. Write the expression “6 more than x” as a variable
expression.
“6 more than x”
x + 6 Identify the variable.
“more than” often indicates an addition.
Look for keywords in expression.
2. Write “12 decreased by b” as a variable expression.
“b decreased by 12”
b – 12
“decreased by” often indicates a subtraction.
3. Write “2 less than a, cubed” as a variable expression.
“2 less than a, cubed”
(a – 2 )3
“less than” often indicates a subtraction.
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
5
Examples: 1. Evaluate “the difference between x and the total of 4
times x and 2” when x = 15.
“The difference between x and the total of 4 times x and 2.”
x – ( 4x + 2 )
Identify keywords.
Identify parts of the phrase
– 3x – 2
Simplify.
that can be grouped on
– 3(15) – 2 Evaluate at x = 15.
their own.
– 47
2. Evaluate “the sum of 4 and y, divided by the
square root of x” when x = 4 and y = 6.
4 y
x
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
4  (6)
10
 5
(4)
2
6
Examples: 1. Find the value of the (4x + 3)2 + |x| when x = – 2.
= (4(– 2) + 3)2 + |(–2)|
= (– 8 + 3)2 + 2
= (– 5)2 + 2
= (– 5 ) • (– 5) + 2
= 25 + 2
= 27
Evaluate expressions within
grouping symbols.
Simplify the exponent.
Add.
3a  b when a = 3 and b = – 1.
a3
3 (3)  (– 1) 8


Division by zero is undefined.
(3)  3
0
2. Evaluate
This expression is undefined when a = 3 and b = – 1.
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
7
Example: Write a variable expression for “A number plus the
product of the number and 5.” Evaluate this expression
when “a number” is 2.
x + (x • 5) Let x = “a number”.
“a number plus” “product of the number and 5.”
(2) + ((2) • 5) Evaluate when x = 2.
(2) + 10
12
Example: Write a variable expression for “There are 6 times as
many cars as trucks.” How many trucks are there if there
are 12 cars?
Let c = the number of cars
c = 6t
(12) = 6t
t=2
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
and t = the number of trucks.
For every truck there are six cars.
Evaluate when c = 12.
There are 2 trucks.
8
4
Example: The volume of a sphere is the product of  and its
3
radius cubed.
What is the volume of a sphere with a radius of 1 meter?
2 meters? 5 meters? Write the answers in cubic meters.
4
Variable expression: V   r 3 Let V = volume of the sphere,
and r = radius.
3
4
4
Radius = 1 m: V   (1) 3    1  4.19 m3
3
3
4
4 
3
Radius = 2 m: V   (2)   8  33.51 m3
3
3
4
4
Radius = 5 m: V   (5) 3    125  523.6 m3
3
3
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
9
Example: To convert a temperature from Fahrenheit to Celsius,
5
subtract 32 and multiply the result by .
9
Convert 72°F to Celsius, and – 40°C to Fahrenheit.
Celsius to Fahrenheit: C  5 ( F  32) Let C = Celsius and
9
F = Fahrenheit.
5
9
Fahrenheit to Celsius: F  C  32 Divide through by 9 ,
5
and add 32 to both sides.
72°F to Celsius:
5
C  ((72)  32)
9
200
C
 22.2
9
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
– 40°C to Fahrenheit:
9
F  (– 40)  32
5
F  72  32  40
10