Download Multiplication and Division Equations 4.10

4.10
Multiplication and Division Equations
Construct Meaning
1
Lynn has traveled 9 2 miles since leaving home to go to a concert.
2
This is 5 of the distance from her house to the concert hall. How
far is the concert hall from her house?
1
2
1
d 9 miles
5
2
Write an equation.
2
19
d 5
2
Simplify numerical expressions.
Convert mixed numbers to improper fractions.
1
2
5
5 19
d 2 5
2 2
1
1
95
3
d 23 miles
4
4
Isolate the variable by multiplying both sides by the multiplicative inverse.
Simplify.
Write each sentence as an equation, solve, and check.
Sentence
Equation and Solution
1
2
1x 2
3
3
One and one-third of a
quantity is the opposite
of two and two-thirds.
4
8
x 3
3
3 4
3
8
x 4 3
4
3
Check
4
8
(2) 3
3
8
8
3
3
x 2
A number divided by
negative six equals
three and five-ninths.
0.7 of a number is
equivalent to the
product of 1.4
and 1.9.
y
5
3
6
9
y
32
6
9
y
32
(6) (6) 6
9
64
1
y or 21
3
3
64
32
(6) 3
9
64
1
32
· 3
6
9
32
32
9
9
0.7z 1.4(1.9)
0.7z 2.66
0.7z
2.66
0.7
0.7
0.7(3.8) 1.4(1.9)
2.66 2.66
z 3.8
Check Understanding
Write true or false.
a. Dividing by a fraction is the same as multiplying by its multiplicative inverse.
2k
k
1
2
3
4
k
1
b. k
c. k d. k k
e. k
3
4
4
3
4
3
5
5
98
Intermediate Course B
Write the multiplicative inverse of each expression if a and b are both integers other than 0.
1
a
2b
f. g. a
h. i. b
b
a
Translate each sentence into an algebraic equation and identify the operation(s) needed to
solve.
j. A number divided by 1.5 is 10.
3
1
k. The product of a number and 3 4 is equal to the sum of 6 and 44.
1
l. One-third of a number is equal to 24.
Did You
Practice
Know?
Write yes or no to indicate whether the value
of x shown is a solution to the given equation.
Equation
Value of x Solution?
1.
3
x 9
5
15
2.
1
4x 12
2
1
3
8
0.5
3. 35.05 7.01x
4.
Indian music also has a complex
rhythmic structure.The player
of the tabla (hand drums) often
must play a different rhythm
with each hand.Western jazz
musicians have adapted a
variety of complicated rhythms
and time signatures from the
music of other cultures.
x
1
3 1
3
8
Write the multiplicative inverse of each number.
2
1
5. 6. 2
7. 17
15
2
1
2
Solve and check. Remember to simplify each side of the equation before isolating the variable.
k
5
1
1
1
15
8. p 5
9. 1c 2
10. 2
6
6
4
7
16
11. 3.2x 4 0.8
1
4
1
4
1
2
14. 4d 2 3
1 1
4 2
3
4
12. t 12
f
6.3
15. 2 3
5
3
4
13. j 3
1
2
1
2
16. m 1 1
Translate each sentence into an algebraic equation and solve.
17. Three and one-half times a number n is the opposite of seven-ninths.
18. A number n divided by three and one-half is the opposite of seven-ninths.
Apply
Write an equation for each word problem. Then find the solution.
19. One year after buying a used car, Isabel sells the car for two-thirds the price she paid
for it. If Isabel’s loss was $825, how much did she pay for the car?
20. Georgia is buying a washing machine for $540. She must make a 20% down payment,
then pay off the balance in six equal monthly installments. How much will she be
paying each month?
21. Riley wants to rent a van that will cost $60 plus $0.59 per mile. If he has budgeted $370
for the rental, what is the maximum number of miles he can travel?
© Copyright 2004
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7.1
Constants, Variables, and Terms
Construct Meaning
In the process of investigating, it is helpful to break down a complex issue into its
components. When you are working with algebraic expressions, it is important to identify
its parts. Identifying the parts of an expression will help you understand the expression
and combine parts correctly.
The variable x can represent
the number of people.
Variables are symbols, usually letters, that represent
different values.
Constants are symbols or letters that represent fixed values. A
constant factor is a coefficient. It is the number by which a
variable is multiplied. A number by itself in an algebraic
expression is also called a constant.
The symbol represents
a constant.
12 is the coefficient of
z in the product 12z.
Terms are numbers, variables, or products of numbers and
variables. In an expression, terms are separated by a plus or
minus sign. Like terms contain the same variables. Matching
variables are raised to the same power.
3x and 5 are terms in
the expression 3x 5.
6y and 2y are like terms.
Identify the parts of the expression 2x3 x 20 x3.
2x 3 x 20 (x 3)
Rewrite subtraction as addition of the opposite.
3
3
2x 1x 20 (1)(x )
The terms are 2x 3, x, 20, and x 3. The like terms are 2x 3 and x 3.
The coefficients are 2, 1, and 1. The constant term is 20.
Operations with Variables and Terms
Any two variables or terms can be multiplied.
x · y xy
z2 · z3 z5
To multiply powers with the
same base, add the exponents.
Like terms can be added or subtracted. Variables do not change when combining terms.
x x 2x
5ab 12ab 7ab
Terms that do not have matching variables cannot be combined into one term.
4x 3y
x2 x3
Simplify each of the following expressions.
12x 20y x
12x 20y (x)
Rewrite subtraction as addition of the opposite.
12x (x) 20y
Apply the Commutative Property.
13x 20y
Add like terms.
4x 3 · (5x 2)
4 · (5) · x 3 · x 2
20 · x 5
20x 5
Apply the Commutative Property.
Multiply the coefficients and multiply the variables.
Simplify.
Be diligent to present yourself approved to God, a worker who does not
need to be ashamed, rightly dividing the word of truth. 2 Timothy 2:15
156
Intermediate Course B
Check Understanding
a. Find the pairs of like terms in the chart.
b. In physics, g always represents acceleration due to
7z
5xy
ab
7z
�5x
6ba
10m2n2
2m2
10m2n 3y 2z 3
p5q3
3z 2y 3
3m2
25x
�q5p3 �9z 3y 2
gravity. This is an example of a
.
2 2
1 2
c. Are the terms �3� y and 1�3� y like terms? Find the
sum and then the product of these terms.
d. How many terms appear in the expression �10m � 2mn � 8m � 6n2?
e. Are constant terms in an expression considered to be like terms?
f. Translate “the sum of a number and fifteen more than the number” into an algebraic
expression and simplify.
Practice
Copy and complete the chart by listing the items in each category.
Expression
1.
5x � 7y � x5 � 2
2.
k
3k � 6 � �� � 10
2
3.
8a2b3 � 2b3a2 � 8
Terms
Like Terms
Coefficients
Constant Terms
State whether the terms in each set are like or not. Find the product and then the sum of
each set of terms. Simplify if possible.
3
1
4. 2m, m
5. 5k, �k
6. 4xy, 4yx
7. 2a, a2
8. ��c3, ��c3
9. 4p2, 2p4
8
8
Simplify each expression.
10. 5a � 2a
2
5
13. 2m2 � ��m � ��m
7
7
16. 4xy � 2xy � 6
11. 7x � x � 7
12. �3c � 8 � c � 12
14. 2d � f � 6d
15. 5z3 � 3z5
17. ��p3 � 4 � ��p3 � 6
18. �3r � 2s � 2r � 2s
4
5
3
5
19. Consider the expression �15x � 20x � 4.
a. Evaluate the expression for x � �3.
b. Simplify the expression and evaluate the simplified expression for x � �3.
c. Compare your answers from a and b. Explain.
Apply
Translate each of the following into an algebraic expression and simplify if possible.
20. The sum of twice a cubed number and the opposite of the cubed number
21. The sum of twice a squared number and the opposite of the cubed number
22. Your friend has 17 more books than you have at home. Write an expression in simplest
form that represents the total number of books you have together.
Explain the use of the italicized word in each sentence.
23. “The constant in the investigation was the time of day of each crime.”
24. “The variable in the investigation was the method used to break into each home.”
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