Algebraic Expressions and Variables. Problem Solving Analyze the problem. Define your variable. Form...
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![Page 1: Algebraic Expressions and Variables. Problem Solving Analyze the problem. Define your variable. Form an equation. Solve the equation Check the solution.](https://reader038.fdocuments.us/reader038/viewer/2022110322/56649d3f5503460f94a19013/html5/thumbnails/1.jpg)
Algebraic Expressions and Variables
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Problem Solving
• Analyze the problem.
• Define your variable.
• Form an equation.
• Solve the equation
• Check the solution
• State the conclusion
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In order to form equations, we need to know how to translate words into mathematical expressions. In this section, we will learn how to do this.
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Definition
division.or tion multiplica
n,subtractio addition, of operations
by the together joined variables
and numbers ofn combinatio aor
number a is Expression AlgebraicAn
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ExampleWhich of the following are expressions?
16 b
1152 x
722 yx
ax 432
1
52 xb
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Translating Phrasesthat involve Addition
• The sum of 2 and x 2 + x
• 8 increased by 2 8 + 2
• 7 more than v v + 7
• 6 greater than x x + 6
• Exceeds L by 20 L + 20
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Translating Phrasesthat involve Subtraction
• The difference of 2 and y 2 - y
• c decreased by 2 c - 2
• 7 less than v v - 7
• 6 reduced by x 6 - x
• 20 less L 20 - L
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Translating Phrasesinvolving Multiplication
52 5 and 2 ofproduct The
xx 10 times01
yy 2 twice
zz 3 triple
xx2
1 number a of
2
1
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Translating Phrasesinvolving Division
520 5 and 20 ofquotient The
4 4by divided
TT
12 parts equal 12 intosplit
xx
12 12 to of ratio Thea
a
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Translating the following phrases:
p and 12 of difference the
ba and ofquotient the
c and 7 ofproduct the
32 of 4
3
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Definition
division.or tion multiplica
n,subtractio addition, of operations
by the together joined variables
and numbers ofn combinatio aor
number a is Expression AlgebraicAn
Recall
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Evaluating Expressions
simplify. toagreement
operations oforder theuse then value,
specified with the variable thereplace
,Expression Algebraican evaluate To
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Example
.6 when 27
32 Evaluate
xx
simplify then 6,by replace We x
27
362
35
15
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Example
5 ,1when
62 Evaluate 2
gh
ggh
simplify. then 5by and 1by replace We gh
256152
23010
160040 2
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Example
2 when 43 Evaluate 2 aaa
simplify. then 2by replace We a
22423
446
22166
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Definition
.quantities moreor two
between iprelationshknown a state
toused are Formulas . variablesmore
or ith twoequation wan is formulaA
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Formula for Sales Price
discount - price original price Sales
formula the
get ediscount w theis and price,
original theis price, sales theis If
d
ps
dps
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Example
price. sales
thefind $22, isdiscount theIf $45. is
shirt a of price original theSuppose
.for solve
then22,by and 45by replace We
s
dp
dps 2245 s
23s$23. is price sales The
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timerate traveledDistance
trd
gone?you havefar How hour.per miles
65at hours 3for driveyou Suppose
3 and 65 case In this tr
365d
195dmiles. 195 traveledhaveYou
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.Conversion eTemperatur
9
325 F-C
.Fahrenheit degreesin re temperatu theis
Celsius degreesin re temperatu theis
F
C
Celsius. degreesin re temperatu theFind
. 104 wasReddingin high today The F
9
321045 C
9
725
9
360 40
C40 wasre temperatuThe
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sExpression Algebraic gSimplifyin
s.expression algebraicsimplify to
tionmultiplica of properties theusecan We
xxx 123434
tion.multiplica theof grouping thechange
tion tomultiplica ofproperty eassociativ
theused weabove example In the
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s.expression following heSimplify t
Examples
6y4
r4-3
w3-9-
s2-7
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s.expression following heSimplify t
More Examples
2842 xxx
s43s
239 tt
yx 27
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The Distributive Property
cabacba
cba
numbers,any are and ,, If
cabacba
cba
numbers,any are and ,, If
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s.expression following heSimplify t
Examples
324 3424 20128
38 x 388 x 248 x
yx 3 yx 33 yx 33
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s.expression following heSimplify t
Examples
544 5444 42016
37 x 377 x 217 x
yx 3 yx 33 yx 33
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.expression following heSimplify t
More Examples
243 x 2343 x
612 x
612 x
435Simplify a
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The Extended Distributive Property
dacabadcba
dcba
numbers,any are and , ,, If
dacabadcba
dcba
numbers,any are and , ,, If
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s.expression following heSimplify t
Examples
zyx 4322
cba 323
zyx 24
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The Right Hand Distributive Property
acabacb
ba
numbers,any are c and ,, If
acabacb
ba
numbers,any are c and ,, If
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s.expression following heSimplify t
Examples
454
32 yx
523 yx
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Terms Like Combining
. variablesmoreor one andnumber
a ofproduct aor number a is A term
terms?are following theofWhich
x2 y73 5
x2 24x abc7
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t.coefficien thecalled
isnumber the, variablesmoreor one
andnumber a ofproduct theis terma If
each term.in t coefficien heIdentify t
yx23- a2
25rt w105
t u4
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exponents.
and variablessame thehave they if
like,or similar be tosaid are Terms
Similar Terms
rms?similar te are following theofWhich 29 and 9 tt xx 7 and 4
xyxy 9 and 2- abba - and 2 2
22 - and 5 xyxy wxywxy 3- and
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BIG DEAL
similar. arethat
ermssubtract tor addonly can You
same. theremainspart
variableThe ts.coefficien hesubtract tor
addsimply rms,similar te combine To
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s.expression following heSimplify t
Examples
xyx 432 aba 734
yxyx 5332
trtr 521325
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s.expression following heSimplify t
More Examples
143 ww baba 75582
nmnm 98 yxyx 25734
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Using the Distributive Property
following. heSimplify t
4253 xx
422533 xx
82153 xx
81523 xx
75 x
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s.expression following heSimplify t
Examples
tt 2936
13264 xx
ww 523475
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Using the Distributive Property
following. heSimplify t
1234 xx
1234 xx
122344 xx
22124 xx
102 x
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Using the Distributive Property
follows. as problem isshorten thcan We
1234 xx
22124 xx
21224 xx
102 x
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aa 1 of opposite The
Multiplying by -1
aa 1
3 x 31 x
311 x 3 x3 x
33- so xx
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s.expression following heSimplify t
Examples
yx 32
ba 35
ts 43
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s.expression following heSimplify t
More Examples
523322 xx
95423 aa
783 xx
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Equations Solving
equations. dcomplicate more solve
can wes,expression dcomplicate more
simplify tohow know that weNow
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equation.
an solve toneeded steps theRecall
equation. theof sideeach Simplify
n.subtractio
or addition by variable theIsolate
division.or
tion multiplicaby equation theSolve
solution. Check the
![Page 47: Algebraic Expressions and Variables. Problem Solving Analyze the problem. Define your variable. Form an equation. Solve the equation Check the solution.](https://reader038.fdocuments.us/reader038/viewer/2022110322/56649d3f5503460f94a19013/html5/thumbnails/47.jpg)
116728
equation theSolve
r
521 r
15121 r
62 r
2
6
2
2
r
3r
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131547
equation theSolve
xx
xx 31447 xxxx 3314347
1477 x
714777 x217 x
7
21
7
7
x
3x
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33247
equation theSolve
tt
66728 tttttt 7667728
628 t66628 t
t22
3232747 tt
66728 tt
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equations following theSolve
575 y 34 c
462514 rr
2462 x 40510 x
372645 xx
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Problem Solving
• Analyze the problem.
• Define your variables.
• Form an equation.
• Solve the equation
• Check the solution
• State the conclusion
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Suppose apples sell for 59 cents per pound. Find the cost of
• Five pounds of apples.
• x pounds of apples.
• M pounds of apples.
• (p - 2) pounds of apples
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Example
Suppose you want to construct a rectangular garden that has a perimeter of 600 feet. If the length is twice the width, what should the dimensions of the garden be?
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Example
After failing your first exam, you study really hard to improve your grade. On the second exam your score is 22 points more than the score on your first exam. If the total of your two exam scores is 144 points, what were your two test scores?
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Example
In a biology course, students spend a total of 250 minutes in lab and lecture each week. The amount of time spent in lab is 50 minutes shorter than the amount of time spent in lecture. How much time is spent in lecture each week?
![Page 56: Algebraic Expressions and Variables. Problem Solving Analyze the problem. Define your variable. Form an equation. Solve the equation Check the solution.](https://reader038.fdocuments.us/reader038/viewer/2022110322/56649d3f5503460f94a19013/html5/thumbnails/56.jpg)
Example
A landscaper buried a water pipe around a rectangular-shaped lawn to serve as a supply line for a sprinkler system. The length of the lawn is 5 times its width. If 240 feet of pipe was used, what is the width of the lawn?
![Page 57: Algebraic Expressions and Variables. Problem Solving Analyze the problem. Define your variable. Form an equation. Solve the equation Check the solution.](https://reader038.fdocuments.us/reader038/viewer/2022110322/56649d3f5503460f94a19013/html5/thumbnails/57.jpg)
Example
In an effort to cut costs, a company decided to lay off 5 employees every month until the number of employees totals 465. If 510 people are now employed, how long will it take to reach the employment goal?
![Page 58: Algebraic Expressions and Variables. Problem Solving Analyze the problem. Define your variable. Form an equation. Solve the equation Check the solution.](https://reader038.fdocuments.us/reader038/viewer/2022110322/56649d3f5503460f94a19013/html5/thumbnails/58.jpg)
Example
A preschool charges $8 per child to attend the morning session, and $10 to attend the afternoon session. No child can attend both. Thirty children are enrolled in the preschool. If the daily receipts are $264, how many children attend each session?
![Page 59: Algebraic Expressions and Variables. Problem Solving Analyze the problem. Define your variable. Form an equation. Solve the equation Check the solution.](https://reader038.fdocuments.us/reader038/viewer/2022110322/56649d3f5503460f94a19013/html5/thumbnails/59.jpg)
Example
At a service station, the underground tank that stores regular gas holds 100 gallons less than the tank that stores premium gas. If the total storage capacity of both tanks is 700 gallons, how much gas does each tank hold?
![Page 60: Algebraic Expressions and Variables. Problem Solving Analyze the problem. Define your variable. Form an equation. Solve the equation Check the solution.](https://reader038.fdocuments.us/reader038/viewer/2022110322/56649d3f5503460f94a19013/html5/thumbnails/60.jpg)
Example
At noon, a passenger ship and a freighter left port traveling in opposite directions. By midnight, the passenger ship was 3 times farther from port than the freighter. If the distance between the two ships at midnight was 84 miles, How far was the freighter from the port?
![Page 61: Algebraic Expressions and Variables. Problem Solving Analyze the problem. Define your variable. Form an equation. Solve the equation Check the solution.](https://reader038.fdocuments.us/reader038/viewer/2022110322/56649d3f5503460f94a19013/html5/thumbnails/61.jpg)
Example
A salesman receives a commission of $3 for every pair of dress shoes he sells. He receives a commission of $2 for every pair of athletic shoes he sells. After selling 9 pairs of shoes in one day, his commission was $24. How many pairs of each type of shoe did he sell?
![Page 62: Algebraic Expressions and Variables. Problem Solving Analyze the problem. Define your variable. Form an equation. Solve the equation Check the solution.](https://reader038.fdocuments.us/reader038/viewer/2022110322/56649d3f5503460f94a19013/html5/thumbnails/62.jpg)
Example
After receiving a score of 95 and 85 on the first two exams in his math class, Joe skips one exam and gets a score of 0. What scores must he get on his last two tests to get a C in the course (70 average)?
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Example
For every problem answered correctly on an exam, 3 points are awarded. For every incorrect answer, 4 point are deducted. In a 10 question test, a student scored 16 points. How many correct and incorrect answers did the student have on the exam.
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Example
A part-time mover’s regular pay rate is $6 per hour. If the work involves going up and down stairs, his rate increases to $9 per hour. In one week he earned $138 and worked 20 hours. How many hours did he work at each rate?
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Example
Suppose 95 people attended a rock concert. Ticket prices were $60 for reserved seats and $40 for general seating. If 30 people bought reserved seats, how much money was received from the sale of the reserved seats and how much was received from the sale of the general seats?
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Example
In order to receive your degree, the college has decided that you must complete 90 units of college credit. Suppose you have already completed 60 units of credit. How long would it take you to satisfy the requirements if you take 10 units per semester? If you want to graduate in two semesters, how many units should you take each semester?
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Example
After receiving their tax refund, a couple split the refund equally. The husband then gave $50 of his refund to charity, leaving him with $70. What was the amount of the tax refund?
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Example
A fitness club has 150 members. Monthly membership fees are $25 for those under 65 and a special senior rate of $15 for those 65 and older. If there are twice as many seniors as regular paying customers, how much does the club receive each month?
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2222 Examples 3
xxxxxx 5
ttttt 4
Recall that exponents are used to indicate repeated multiplication.
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42 33 333333 63
32 xx 5xxxxxx
Simplify the following:
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nmnm xxx
The Product Rule for Exponents
If x is any number and m and n are
integers, then
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76 55 53 aa
mm 5
94 ss
42 xxx 425 ccc
sss 28
32 ba
Simplify the following:
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232 33 22 62
34x 12444 xxxx
Simplify the following:
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nmnm xx
The Power Rule for Exponents
If x is any number and m and n are
integers, then
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465
73a
25m
94s
4352 xx
4273 aa
342xx
Simplify the following
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2ba baba 22ba
3yx
Simplify the following
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mmm yxyx
The Power Rule for Products
If x and y are any numbers and m is
any integer, then
![Page 78: Algebraic Expressions and Variables. Problem Solving Analyze the problem. Define your variable. Form an equation. Solve the equation Check the solution.](https://reader038.fdocuments.us/reader038/viewer/2022110322/56649d3f5503460f94a19013/html5/thumbnails/78.jpg)
425
7ba
2nm
9ts
532 yx
723 ba
354 sr
Simplify the following:
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Any Questions???