algebraic expressions and variables. problem solving analyze the problem. define your variable. form...

Algebraic Expressions and Variables

Problem Solving

• Analyze the problem.

• Define your variable.

• Form an equation.

• Solve the equation

• Check the solution

• State the conclusion

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.

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

ExampleWhich of the following are expressions?

16 b

1152 x

722 yx

ax 432

1

52 xb

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

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

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

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

Translating the following phrases:

p and 12 of difference the

ba and ofquotient the

c and 7 ofproduct the

32 of 4

3

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

Evaluating Expressions

simplify. toagreement

operations oforder theuse then value,

specified with the variable thereplace

,Expression Algebraican evaluate To

Example

.6 when 27

32 Evaluate

xx

simplify then 6,by replace We x

27

362

35

15

Example

5 ,1when

62 Evaluate 2

gh

ggh

simplify. then 5by and 1by replace We gh

256152

23010

160040 2

Example

2 when 43 Evaluate 2 aaa

simplify. then 2by replace We a

22423

446

22166

Definition

.quantities moreor two

between iprelationshknown a state

toused are Formulas . variablesmore

or ith twoequation wan is formulaA

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

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

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

.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

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

s.expression following heSimplify t

Examples

6y4

r4-3

w3-9-

s2-7

s.expression following heSimplify t

More Examples

2842 xxx

s43s

239 tt

yx 27

The Distributive Property

cabacba

cba

numbers,any are and ,, If

cabacba

cba

numbers,any are and ,, If

s.expression following heSimplify t

Examples

324 3424 20128

38 x 388 x 248 x

yx 3 yx 33 yx 33

s.expression following heSimplify t

Examples

544 5444 42016

37 x 377 x 217 x

yx 3 yx 33 yx 33

.expression following heSimplify t

More Examples

243 x 2343 x

612 x

612 x

435Simplify a

The Extended Distributive Property

dacabadcba

dcba

numbers,any are and , ,, If

dacabadcba

dcba

numbers,any are and , ,, If

s.expression following heSimplify t

Examples

zyx 4322

cba 323

zyx 24

The Right Hand Distributive Property

acabacb

ba

numbers,any are c and ,, If

acabacb

ba

numbers,any are c and ,, If

s.expression following heSimplify t

Examples

454

32 yx

523 yx

Terms Like Combining

. variablesmoreor one andnumber

a ofproduct aor number a is A term

terms?are following theofWhich

x2 y73 5

x2 24x abc7

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

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

BIG DEAL

similar. arethat

ermssubtract tor addonly can You

same. theremainspart

variableThe ts.coefficien hesubtract tor

addsimply rms,similar te combine To

s.expression following heSimplify t

Examples

xyx 432 aba 734

yxyx 5332

trtr 521325

s.expression following heSimplify t

More Examples

143 ww baba 75582

nmnm 98 yxyx 25734

Using the Distributive Property

following. heSimplify t

4253 xx

422533 xx

82153 xx

81523 xx

75 x

s.expression following heSimplify t

Examples

tt 2936

13264 xx

ww 523475

Using the Distributive Property

following. heSimplify t

1234 xx

1234 xx

122344 xx

22124 xx

102 x

Using the Distributive Property

follows. as problem isshorten thcan We

1234 xx

22124 xx

21224 xx

102 x

aa 1 of opposite The

Multiplying by -1

aa 1

3 x 31 x

311 x 3 x3 x

33- so xx

s.expression following heSimplify t

Examples

yx 32

ba 35

ts 43

s.expression following heSimplify t

More Examples

523322 xx

95423 aa

783 xx

Equations Solving

equations. dcomplicate more solve

can wes,expression dcomplicate more

simplify tohow know that weNow

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

116728

equation theSolve

r

521 r

15121 r

62 r

2

6

2

2

r

3r

131547

equation theSolve

xx

xx 31447 xxxx 3314347

1477 x

714777 x217 x

7

21

7

7

x

3x

33247

equation theSolve

tt

66728 tttttt 7667728

628 t66628 t

t22

3232747 tt

66728 tt

equations following theSolve

575 y 34 c

462514 rr

2462 x 40510 x

372645 xx

Problem Solving

• Analyze the problem.

• Define your variables.

• Form an equation.

• Solve the equation

• Check the solution

• State the conclusion

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

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?

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?

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?

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?

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?

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?

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?

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?

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?

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)?

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.

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?

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?

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?

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?

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?

2222 Examples 3

xxxxxx 5

ttttt 4

Recall that exponents are used to indicate repeated multiplication.

42 33 333333 63

32 xx 5xxxxxx

Simplify the following:

nmnm xxx

The Product Rule for Exponents

If x is any number and m and n are

integers, then

76 55 53 aa

mm 5

94 ss

42 xxx 425 ccc

sss 28

32 ba

Simplify the following:

232 33 22 62

34x 12444 xxxx

Simplify the following:

nmnm xx

The Power Rule for Exponents

If x is any number and m and n are

integers, then

465

73a

25m

94s

4352 xx

4273 aa

342xx

Simplify the following

2ba baba 22ba

3yx

Simplify the following

mmm yxyx

The Power Rule for Products

If x and y are any numbers and m is

any integer, then

425

7ba

2nm

9ts

532 yx

723 ba

354 sr

Simplify the following:

Any Questions???