2.1 Evaluating Variable Expressions (Page 1 of 14)
2.1.1
Evaluating Variable Expressions
Variable, Term and Coefficient A variable is a letter used to represent a quantity that is unknown, or a quantity that can change or vary. e.g.
x = the price of one share of Microsoft stock y = the cost of a new car h = a student’s height m = miles per gallon of an automobile
The terms of a variable expression are the addends of the expression. a.
Consider !2x 3 ! 4xy + 2y 2 + 7
Rewrite the expression using only addends (terms).
b. List the variable terms. c.
List the constant term.
d. What are factors? e.
How many factors are in the first term? List them.
f.
How many factors are in the second term? List them.
The numerical coefficients of an expression, or simply the coefficients, are the numerical factors of the variable terms. g. List the coefficients in !2x 3 ! 4xy + 2y 2 + 7
2.1 Evaluating Variable Expressions (Page 2 of 14)
Evaluating a Variable Expression To evaluate a variable expression means to replace the variables with numbers and perform all indicated operations (addition, subtraction, multiplication, division, etc); the result will be a single number. Rule on Substituting Negative Numbers into Variables Whenever a negative number is substituted into a variable, the negative number must be placed inside a set of parentheses. Furthermore, if a negative number is raised to an exponent, the negative number must be inside a set of parentheses and the exponent must be outside the parentheses. Example 1 a. x = !7
Evaluate x 2 when b. x = !4
c. x = 5
Example 2 a. b = !4
Evaluate !b 2 when b. b = !7
c. b = 5
Example 3 Evaluate !y 2 ! 3xy when a. x = !4 and y = 2 b. x = !5 and y = !3
2.1 Evaluating Variable Expressions (Page 3 of 14)
a2 + b2 Example 4 Evaluate when a!b a. a = 3 and b = !4 b. a = !3 and b = !5 .
Example 5 Evaluate x 2 ! 2(x + y) ! z 3 when a. x = 2 , y = !2 , z = !3 b. x = 2 , y = !2 , z = !3
Example 6 Find the volume of a right circular cylinder that has radius 1.25 in. and a height of 5.25 in. Round to the nearest tenth of a 1 cubic inch. Use V = ! r 2 h and 3 ! = 3.14 .
2.2 Properties of Real Numbers (Page 4 of 14)
2.2.1 Properties of Real Numbers If a, b, and c are real numbers, then Property Commutative means order does not matter
Associative means grouping does not matter
Inverse The additive inverse is the opposite. The multiplicative inverse is the reciprocal. Zero
One
Operation Addition Multiplication Commutative property of Commutative property of addition multiplication
Associative property of addition
Associative property of multiplication
Inverse property of addition
Inverse property of multiplication
Addition property of zero
Multiplication property of zero
Multiplication property of one
2.2 Properties of Real Numbers (Page 5 of 14)
The Distributive Property For any real numbers a, b, and c a(b + c) = ab + ac and (b + c)a = ab + ac Example 1 a. !3(4 + 5)
Rewrite each expression using the distributive property. Then evaluate the expression. b. (3 + 5)! 4
solution
!3(4 + 5) = !3" 4 + (!3)" 5 = !12 + (!15) = !27 Example 2 Identify the property that justifies each statement. a. 12 + (!12) = 0 b.
(3 + 11) + 15 = 3 + (11+ 15)
c.
(4 ! 7)! 5 = (7 ! 4)! 5
d.
Any number times its reciprocal equals one.
e.
Any number times zero is zero.
f.
The sum of any number and its additive inverse is zero.
g.
The order in which two numbers are multiplied does not matter.
2.2 Properties of Real Numbers (Page 6 of 14)
Example 3 Create an example that shows division does not have the commutative property.
Example 4 Create an example that shows subtraction does not have the associative property.
2.2.2 Combining Like Terms Terms are addends. Like Terms of an expression are terms that have identical variable parts. Constant terms are always like terms. To combine like terms means to add the coefficients of like terms, keeping the variable part the same. This is possible because of the distributive property (see example).
Example 5
3y + 4 y = (3 + 4)y
a. 4x + 7x
Use the distributive property to simplify each expression. b. 22a + 31a
Example 6 a. 18b ! 4b
Simplify each expression. b. !6t ! 5t
c. 4x ! 13 + 5y ! 3x + 12 ! 2y
= (7)y = 7y
d. 3a 2 ! 2a + 5 + a 2 + 16
2.2 Properties of Real Numbers (Page 7 of 14)
2.2.3 Use the Properties of Multiplication to Simplify Variable Expressions Example 7 a. !4(c " 7)
Use the Commutative and Associative Properties of Multiplication to simplify each expression.
!4(c " 7) = !4(7c) = (!4 " 7)c = !28c b. 2(5x)
Example 8 a. 4(!x)
Commutative property of multiplication Associative property of multiplication
c. 2[a(!15)]
Simplify each expression b.
!14(!5c)
c.
3" 2 % $# ! y '& 2 3
d.
5 ! (!36a 2 ) 6
e.
16(3x)
f.
!5a(!7)
2.2 Properties of Real Numbers (Page 8 of 14)
2.2.4
Simplify Variable Expressions using the Distributive Property
Example 9 Simplify each expression. a. b. 3(4x + 5)
2(5x + 9)
c.
!2(3x ! 7)
d.
!7(2x ! 5)
e.
!3(!4 ! 6x)
f.
!(!3x ! 10)
g.
(3y ! 4)7
h.
(!2x ! 8) 6
i.
4(6b 2 ! 4b + 7)
j.
5(3x 3 ! 7x + 9)
2.2 Properties of Real Numbers (Page 9 of 14)
2.2.5 Simplify General Expressions Steps to Simplify General Expressions 1. Use the Distributive Property to remove any grouping symbols starting from the innermost set and working outward. 2. Combine like terms. 3. Write the terms in alphabetical order with the constant term last. Example 10 Simplify each expression. a. 4(x ! y) ! 2(2x + 5y) b. !7(a ! 4b) ! 2(!2a ! 3b)
c. 3(!2x + y) ! (!3x ! 4y)
d. 3(2a ! 9b) + (3a + 4b)
e. 5x ! 3[2x + 6(!x + 7)]
f. 3a ! 6[a ! 5(2 ! 4a)]
2.3 Translating Verbal Expressions (Page 10 of 14)
2.3
Translating Verbal Expressions into Variable Expressions
In the expressions below marked with an asterisk (*), the actual operation (+, !, ", ÷ ) occurs at the word “and” in the sentence. Verbal Expression
Addition Phrases
Subtraction Phrases
Multiplication Phrases
Division Phrases Powers
6 added to y 8 more than x *the sum of x and z t increased by 9 *the total of 5 and y x minus two seven less than t 5 subtracted from d *the difference between y and 4 m decreased by 3 10 times t one-half of x *the product of y and z 11 multiplied by y twice n x divided by 12 the ratio of t to nine *the quotient of y and z the square of x the cube of z
Variable Expression
y+6 x+8 x+z t+9 5+y x!2 t!7 d!5 y!4
m!3 10t 1 2 x yz 11y 2n x 12 t 9 y z 2
x z3
2.3 Translating Verbal Expressions (Page 11 of 14)
Example 1 Translate each into a variable expression. a. The total of five times b and c. b. Five times the total of b and c. c.
The quotient of eight less than n and fourteen.
d. Thirteen more than the sum of seven and the square of x. e.
Eighteen less than the cube of x.
f.
Eighteen less the square of x
g. y decreased by the sum of z and nine. h. The difference between the square of q and the sum of r and t.
2.3 Translating Verbal Expressions (Page 12 of 14)
2.3.2
Translate into a variable expression
Example 2 a.
Translate “a number multiplied by the total of six and the cube of the number” into a variable expression.
b. Translate “a number added to the product of five and the square of the number” into a variable expression.
c.
Translate “the quotient of twice a number and the difference between the number and twenty” into a variable expression.
d. Translate “the product of three and the sum of seven and twice a number” into a variable expression.
2.3 Translating Verbal Expressions (Page 13 of 14)
2.3.3 Translate into a variable expression, and then simplify the variable expression. Example 3 Translate and simplify each expression. a.
Translate and simplify “the total of four times a number and twice the difference between the number and eight.”
b. Translate and simplify “a number minus the difference between twice the number and seventeen.”
c.
Translate and simplify “the difference between five eighths of a number and two thirds of the number.”
d. Translate and simplify “the sum of three fourths of a number and one fifth of the number.”
2.3 Translating Verbal Expressions (Page 14 of 14)
2.3.4
Defining Variables
Example 4 For each of the following (1) define a variable, and (2) express all unknown quantities in terms of that variable. a. The length of a swimming pool is 20 feet longer than the width. Express the length of the pool in terms of the width.
b. An older computer takes twice the amount of time to process data as a new computer. Express the amount of time it takes the older computer to process the data in terms of the amount of time it would take the new computer.
c.
A guitar string 6 ft long was cut into two pieces. Use one variable to express the lengths of the two pieces.
d. If the sum of two numbers is 10. Use one variable to express each number. e.
An investor divided $5000 into two accounts, one a mutual fund and the other a money market fund. Use one variable to express the amounts invested in each account.
